By being loyal and cooperative citizens, making sure to have an inside traitor, making their plans telepathically, being aware of the security features of the state, and by removing them only in the terminal phases of the plan.

Most of the town-centric plot will be subtle power plays between the "three factions", defector stories from the police force (see the Lives of Others for the small mercies of the secret police), agents provocateur that believe what they're selling, and/or Edward Snowden's story sans media attention. The town can fade into the background, or if the party becomes engaged in quests for one faction or the other, they can start becoming enmeshed into the towns politics.

A discussion of the town's setup can be found here.

Any treason will have to open by excising the necrotic cysts that will inevitably be living in the characters while within the city before the next hindsight investigation. Therefore, this surgery will be the penultimate act of treason before any given coup succeeds.

The motivations for treason will be discussed below, but in short, any effective action will require NPC sponsorship. Either by the Kingdoms of Core Classes, by the Republic of the Sourcebooks financing a coup in its client/buffer state, or via factions within the city itself.

In this answer, I will presume a greater game world populated by two superpowers, the Kingdoms of the Core Classes, and the Soviet Republic of Sourcebooks. (Obviously the names will be rather more in keeping with the world, but...) These mainly have come about due to how the class breakdown needed for the spells to support the cyst state broke down. The details don't really matter before campaign creation, but it's a cute concept, so I'm keeping it for this answer.

The spells themselves rely on security by obscurity. Knowledge of their use is sufficient to employ countermeasures like mind blank or DC 20 heal checks. Actually performing the countermeasures is a death sentence, and so should only be done in the terminal phases of a project. Until then, mind bond or permanent telepathic links are the name of the planning game.

Some literature on defection

The idea of treason has a number of models. In this answer, I'll refer to the model articulated here:

Our research suggests that nearly all motivations can be grouped into four categories - money, ideology, ingratiation, and disgruntlement.

Functionally speaking, we can model defection from the true objectives of ruling the city in the same model, at least as it applies to character plots.

Money

Taylor and Snow discuss money as a motivation:

When linked with secondary factors (disgruntlement, ideology, etc.) money was the motive in 62.6 per cent of the 139 cases ... Until military pay raises in the early 1980s, pay for low ranking military personnel was traditionally meager and many of our cases turned to espionage merely to pay outstanding debts or to allow them to live more stylishly.13 Rather than turn to family, friends, or banks for loans, they turned to the KGB or related services.

The thing about this cystarchy is that it's a remarkably effective communism, almost by accident. The basic necessities are provided by magic (The Wall of stone, fabricate, and sustain spells respectively), and taxes are gathered in the form of XP that anyone can donate effectively via good, evil, or (if houseruled) neutral acts.

Money, is therefore, going to be a problem. Private debts on things the state disapproves of, luxury goods beyond one's means, extra-marital affairs, and every other private act that society may disapprove of requires resources. This is a powerful dramatic motivator for treason, and therefore a good motivation for an NPC to bring the party in on the "secret." Moreover, money can also map to "power", and thereby provide power struggle motivations for even the rulers.

Ideology

Taylor and Snow discuss ideology as a motivation:

In fact, all traitors, regardless of national origin, have reacted to somewhat similar changing social circumstances. Many early traitors in the West appeared to be motivated by ideological considerations - at least to the degree that anyone acts on single motivation.

From contemporary media, the pattern of Edward Snowden is most obvious here. We will presume a character employed in the secret police called Eddard Frostfell, just because. Ideological treason suggests a memetic contamination, that the "other side" somehow has a much better weltanschauung, and that the state currently employing the NPC should be fixed to more accurately resemble the desired outcome.

The character, involved in acts that are not supported by his/her ideology, wants to fix the system/city/world. Given that this implicit campaign setting has strong ideological undertones, memetic contamination from "the old hit die empire" (whatever the city is sitting on as a leftover from the great wars), or from "the kingdoms of the core classes" can certainly inspire an NPC to wrongly desire freedom of thought and of speech, and to involve the PCs in a plot to broadcast the rightful controls of the state to a world that will actually rather care about them.

Ingratiation

Taylor and Snow on ingratiation:

Ingratiation becomes a motive when information is betrayed to foreign sources in order to fulfill friendship or love obligations or in order to make favorable impressions on someone whose approval is desired. ... The use of same- or opposite-sex 'honey traps' to attract and, if necessary, blackmail potential traitors has been a favorite theme of novelists and scriptwriters, but has been little understood and vastly overrated by them

Discuss this with your players at campaign start, as it may broach taboo topics at the game table. This is likely the most effective "external" motivation to treason, and the characters can be sent in with this mission in mind. It's an excellent counterbalance to the mostly internal motivations of the prior two reasons, and the setups for blackmail, friendship, or honey traps can be quite fascinating from an intrigue perspective.

Disgruntlement

Taylor and Snow, again:

Like money, disgruntlement is a motive of growing importance. By disgruntlement we refer to the sense of personal dissatisfaction that stems from feelings of being overlooked, overworked, and under-appreciated. ... People entrusted by their government with classified information are given a unique ability to exact revenge on that government if it offends them

The most prosaic of motives, this can provide motivation for any NPC in the city to try to recruit the characters, terroristic actions or sabotage either for heightening the character's importance (expect a trap), or for revenge for a slight, This most egoistic of motives can provide fine gaps in the armor of Truth, Comradeship, and Surveillance.

Countermeasures and unexpected threats

A perfect system is boring, since it cannot be threatened usefully. However, no system is perfect, especially when modeled on the post-war soviet republics. Based on the casting needs of the city, and the heritage of the soviet pattern we're copying, a triumvirate makes sense for real control of the city. In the soviet republics, it was the Party, the Army, and the KGB (with a private tussle on the side between the KGB and the GRU.) The party, here, maps to the ideologue. The army, here, maps to Mr. Mayor, and the KGB maps to Judge Fred. In the smaller client states, less is trivially researchable, and more details here would just turn this seed of a campaign setting into a campaign setting. Here's some light reading on the balance of power in East Germany.

Clearing up misconceptions, or where the town isn't vulnerable

Unfortunately, for the players, any approach at attacking the town must be made with intention, research, and subtlety. Direct confrontation of the town, absent betrayal from inside (one way or another) will lead to the "joyful cooperation" and subsequent death of the adventuring party. Most groups have a sledgehammer approach to espionage and diplomacy. Make it clear to the group that this city is a tough nut to crack, but at the same time, repays loyalty for loyalty.

More to the point, the city is designed with adventurers in mind. There are always quests and external threats available, simply because adventurers are useful idiots who, if not kept amused, will try to figure out what's going on in the town.

Mid-level direct force assaults, if the adventurers have citizenship or "extended visas" will result in instant detonation via necrotic bloat (no save, no sr, just vile damage) or worse.

Where the town is vulnerable: The triumverate

At the end of the day, the town is most vulnerable when the triumvirate are playing "games" with each other. Each member of the triumvirate is responsible for a different aspect of the city's defenses, and can therefore weaken those defenses to support their own assets.

Mr. Mayor's play

The mayor is responsible for the army, and for citizen "trend-finding" via his feasts and mindsight. He is the popular figure and the titular leader. While he is not immune from espionage from the other factions, most of his decisions do not require explanation. Therefore, he is the quest giver.

All level parties will get quests from him, all of which have (at the least) colourable excuses why they benefit the city. Most actually do. The subset that do not are busywork intended to keep adventurers busy. However, as parties level up to a useful level, the mayor can start arranging for private lines of communication (sending and whatnot) which are completely justified in the context of the quests given.

He can set up external caches of equipment, and has the easiest time getting external resources in play. At the same time, since he's at the top, his main game will be countering the power plays by the other members of the triumverate, the sponsor state of the soviet republic of splatbooks, and the kingdoms of core classes. The danger is if the pawns start thinking for themselves when they are finessing a power play for the opposition, everything can start exploding.

Judge Fred's play

She is subtle, quiet, and quite a nice person (in image.) She doesn't have to be feared because she can dominate anyone who gives her sass. (Or detonate their brains if mildly irked.) As Judge, Jury, and corrections officer, Judge Fred has astonishingly high powers. As the ultimate recipient of Hindsight and the leader of the police forces, her power inside the city is almost completely absolute.

Which is why, if she wants to overlook a plot or two, so long as the plots do not provide actionable intelligence to the ideologue's Susurrus of the City, Identify Transgressor, and Expose the Dead forensic work, it is completely within her power. As she also is in charge of the Agents Provocateur, (and the secret leader of the thieves guild, and the rebel leader in the city, etc. etc.) she is in a position to instigate as many plots as she likes, so long as she doesn't publically get egg on her face.

Her primary interactions with the party is as corrections officer, turning them into informants and useful pawns when they inevitably do something naughty. She can bypass the border protections of the town trivially, and arrange for unexpectedly little opposition in key locations during power plays.

Of course, since all of this can backfire on her, most of her agents are wired to multiply-backuped save-or-die contingency spells. Since she deals with public order, all but the most spectacular FUBARs of her plots can be contained by her. Which just leaves situation normal and "everything's exploding."

The patron saint of "lazy" parties, the party who takes her blessing will find that it's unwise to depart from her rails. In terms of planning and capability, think Keyser Sose. She never appears in her person to assets she's managing (police or otherwise), leaving layers of either handlers, or illusions to confuse the issue. There are no public trials, as offenders are silently dominated, escorted to a room where they can plead to an empty room for mitigating circumstances, be thoroughly mind-probed, and then committed to the tax office for joyful correction. And they will be happy. By order.

The Ideologue's play.

The ideologue is a public academic, and private kingmaker. Having control of "the party" (soviet party not PC party) line, all respectable citizens are encouraged to voluntarily join the party (in a very soviet fashion). Having the job of determining "rightness" and "correctness", her power is more abstract than the other two, but as she can overwhelm the other two with a mob of right-thinking citizens, or call in abstract support from the parent state, angering her is severely contraindicated.

She is the most likely to start an actual power play to depose Mr. Mayor in favour of a new "candidate", either from the parent republic or from her own volition. Her benefits are, in fact, those of ideology. She knows a huge amount of what is correct, and can therefore instruct her agents how to act impeccably. Hers is the easiest job of defeating hindsight, as she can arrange matters such that nothing is actually suspicious until it's too late.

The patron saint of parties who really want to overturn the city, she can show them how to act correctly until it's too late. The only trick will be making the initial contact and trust. She'll likely use oversight on Judge Fred's hindsight collection summary to find promising candidates, or have promising candidates sent from the soviet republic of splatbooks (which is an excellent campaign premise.)

The danger of her approach is that once people are instructed in right-action, so long as they do not have cause to be mind probed by Judge Fred or otherwise trigger any alarms, they can pass unnoticed in the daily surveillance feeds.

Mid-level agents provocateur

Borrowing from the story of Ulfric Stormcloak, (and ignoring any historical parallels for fear of invoking Godwin's law), the leaders of rebellions often have strong ties to the opposition's intelligence apparat. If one is groomed as an Agent Provocateur, it doesn't take too much to slip the leash and actually believe in the cause one is peddling. Since you're expected to be doing the things that you're doing anyways, exposing real idiots to fill your quota, and reserving the useful idiots to actually accomplish things for one's cause is a time honoured technique that has blown up in more than one intelligence agency's face.

External regimes and defecting secret police

Defecting secret police can provide a plot hook to an external party to investigate the town properly. So long as the party is sufficiently subtle about it, and that they're aware of the dangers that they face, the defecting policeman can start off the campaign's true plot arc. Much of the city's security, unfortunately, requires that bad actors aren't aware of it (tsk. tsk.), and even the glimmer of an idea is enough to start unravelling the threads for a sufficiently competent party. (Make sure the players know that TPKs are on the line when they're in the city.)

External regimes operate in much the same way, save for the fact that they provide a mission structure and resourcing as well as motivation.

Dealing with the damned cysts

The greatest danger of the necrotic cyst is when it's buried deeply without notice. The second greatest danger is that people in the city without cysts are immediately marked as foreigners. If there is evidence of cyst removal (i.e. no visa), no mercy will be provided as the party is escorted to the joyful corrections office.

Extracting them, however, is simple. Once identified, a DC 20 heal check can extract the thing (don't fail, or the character dies), and the character has until the next intelligence sweep to get out of the city or execute her plot.

Dealing with hindsight

Hindsight is a very specific spell. It shows actions and conversations. So long as a plausible cover conversation or action is being maintained, there is no evidence of telepathic communication. (Which doesn't prevent detection from other sources, but none are as comprehensive and merciless as hindsight.)

The trick here, in the city, is don't be suspicious. Mind blank will foil the spell, as:

In the case of scrying that scans an area the creature is in, such as arcane eye, the spell works but the creature simply isn’t detected.

but this requires perfect secrecy, as the moment that any evidence of presence is detected (conversations directed at the air, doors opening without cause), a massive man hunt will be on, and non-divinatory signals are still possible. Still, the spell itself can be foiled without too much trouble in the end-stages of a plot.