You actually have a built in answer in your question.
"...she wrote that she is seeking out her patron so that it can help her fly over some ancient city walls."
When you couple that with:
"I explained to the player that fly is already a warlock spell that she will have access to at higher levels..."
Now, you want to know how to do that in game? Take on the role of her patron so that her patron explains to her that she WILL be able to fly over the walls as she desires, it will just require some more power. This is an excellent RP scenario for you to engage in with your player. As for hunter's mark, when your player starts talking about ways to single out foes and weaken them, the Patron should be bringing up the Warlock version of Hunter's Mark, which is called Hex.
As for the other classes, it's really up to them. Here's my personal breakdown, logically, of how they know about what spells they have access to:
- Wizards - Through meticulous study of books and practical exercise.
- Arcane Tricksters - Same as wizards (PHB covers this under the class)
- Eldritch Knights - Same as wizards (PHB covers this under the class)
- Bards - Through the colleges they attend. Their knowledge of magic can be either refined, or broad, based on their college of study.
- Sorcerers - Since their power originates within them, I personally feel that they can sense the magics available to be tapped in the weave because they have a primal connection to it.
- Warlocks - Their knowledge comes from their other-worldy patron. The warlock class has some access to casting, but more access to Invocations which act more akin to spell like abilities. They learn about their power from their actual source and have struck some kind of bargain to tap into it.
- Divine Casters - This is Paladins, Rangers, Druids and Clerics. They pray to their gods. As they increase in power and knowledge of the world (level up), their gods grant them further insight into the Divine domains.
The Way of the Four Elements Monks aren't on the list above because they aren't actually spell casters. They have Discipline of the Elements instead, which functions very differently from spell casting in that it uses Ki points rather than spell slots. In addition, you select from a list of disciplines, not from a spell list.
But remember, that's my personal take on the source of magical power and knowledge that's only kind of backed up by the books in the sense of the Weave and the Divine. What's really great about the D&D multiverse is that you don't need to subscribe to the concept of the Weave and the Divine at all. You can play a game in our world, where magic is granted by the Egyptian Pantheon. Or you can play in a Final Fantasy setting where magic comes from sources like materia. Or you can tap other materials as sources of magic, like dragon souls, energy fonts, sheer willpower, or radiation.
The possibilities are limitless. It's up to you, and your players, to describe how they know about certain things and whether or not that makes sense consistently within your game world.
Based on the stats given, it appears to me that your best option is to not multiclass.
The most obvious option would be a dip into Monk, to get Unarmored Defense's Wis-to-AC bonus, but that would be trading leather's +1 for your Wisdom's +1. The other abilities wouldn't really add anything to what you already have.
Another option is Barbarian's Unarmored Defense. That would add +2 AC from your Con instead of your Leather armor, but simply changing to Studded Leather (also light, and giving full dex bonus) would achieve the same result. The Barbarian's Rage does add a moderately useful ability, but comes at the expense of delaying your progressing in Fighter/Monster Hunter.
Finally, a 2 level dip into ranger would allow you to pick a second fighting style, specifically "Defense" which grants +1 AC while wearing armor. Again, the Ranger's abilities do not synergize particularly with your other abilities, and delaying your main class abilities for 2 more levels is unlikely to be worth +1 AC.
Also note that none of these options combine. Unarmored Defense is specifically called out under multiclassing as an ability you can only get from a single class, and even the greatest Cheeselords still have trouble finding a way to be unarmored and wear armor at the same time.
It seems like you already have the best option figured out: take the Dual Wielder feat for the bonus AC, and indirect damage increase of 2 one-handed weapons.
While you're shopping for a second rapier, upgrade to a suit of studded leather. This and Dual Wielder will bring you up to 17 AC.
After that, just stick with fighter. At 7th level you can start using 2 superiority dice on damage, as well as automatically maxing superiority dice damage against certain types of enemies. At 10th level, your superiority dice get bigger by a step, becoming d10's.
These options will do more for you than you can get from any quick dips into other classes.
For a build like yours, and for most builds in 5e, multiclassing is a great way to increase the breadth of your skills and abilities, but usually does not add a lot to the depth of them.
As Ethan has pointed out in the comments, don't forget to turn to your teammates for assistance and buffs, as well as doing the same for them where you can.
This is known as a "re-spec", and it's reasonable on occasion.
While changing one's character levels isn't a standard rule in D&D 5th edition, it's not unreasonable to allow a character to change a poorly-built character on a one-off basis like this. In fact, the Adventurer's Guild organized play rules explicitly allow low-level characters to do this:
This is commonly known as a "re-spec" or "rebuild", and I've seen it used well in the first D&D campaign I ever played in. Most of our gaming group was new to D&D at this time, and many players made mechanically very sub-optimal choices as a result. When we reached level 10, the DM allowed us to change our characters once only, to ensure that nobody would be laboured with an ineffective character due to poor decision-making early on.
It's not unbalanced, since a re-spec is fundamentally no more powerful than a normal character of their level. It's mechanically identical to allowing the player to select a new character of the same level (something many DMs, myself included, allow when a character dies or retires), except that this way they retain their existing character identity and story, which is usually a good thing. And while AL currently only allows it up to level 4 (as noted by nick012000), you aren't bound to that rule unless you're playing in Adventurer's League.
However, I recommend it be allowed very sparingly, such as once per campaign. Allowing frequent re-specs would give characters too much flexibility (e.g. completely changing their character to suit each adventure or scenario), which would lead to an effective increase in power. It may also undermine the continuity of the game, and lead to much time being spent in character generation instead of playing the actual game.
You may like to offer the other players the chance to re-spec at the same time, just to be fair. However, again, it must be made clear that this is a one-off grace which players will not expect to be repeated. As DM, you are within your authority to bend the rules to accomodate players, but don't let them talk you into doing this often, or the rules lose meaning.
Youtuber and excellent Dungeon Master Matt Colville, of Running the Game fame, has in the past stated that he allows his players a great-deal of leeway in re-spec. He considers it important that characters get to play the sort of character they want to play, and that if the player's character concept doesn't match what they had mind when they made their decisions, that they should be able to change it. In one case he recently allowed a character to change a spell mid-session when it was discovered to be less effective than the player had assumed when they made that choice.