Glad you could reverse-engineer the save DC calculation. It is actually described in the SRD - and in several other places. Kept that one as an example.
Step 8: Special Abilities and Qualities
Most special abilities that cause damage, such as breath weapons, give a save (Fortitude, Reflex, or Will depending on the ability). The DC for almost all special abilities is equal to 10 + 1/2 the creature’s Hit Dice + a relevant ability modifier (usually Constitution or Charisma depending on the ability). Special abilities that add to melee and ranged attacks generally do not allow a save, as they rely on the attacks hitting to be useful.
So a special attack that doesn't already rely on a hit roll will have a save DC. Some special attacks that are particularly devastating might have both hit rolls and saving throw, but those are rare.
As Geoffrey points out in his answer, some spell-like abilities that emulate published spells don't use the 1/2 HD to calculate the save DC, instead using the spell level of the emulated spell for the DC calculation. Some unique (Sp) abilities still use 1/2 HD though.
Finally, Supernatural (SU) abilities ignore spell resistance, but spell-like (SP) abilities usually need a caster level (almost aways equal to HD, unless otherwise specified) to beat SR.
Estimate it
One rule that a lot of DMs forget is that not every single combat encounter—or more specifically, every single moment within combat—needs to be handled in the Combat-Initiative Abstraction. A group of level 5 characters facing off against a single CR 1/4 goblin doesn't need to open with Initiative, and similarly, if a group of characters are up against any encounter whose difficulty is trivial to surmount (or they've whittled an encounter down to this point), you shouldn't feel compelled to mechanically iterate through each moment of that combat. Just use a single d20 roll (maybe a contested ability check?) to determine how many hit points are used up.
Surrender!
It depends on what kinds of creatures the players are up against, but most NPC creatures possess a modicum of intelligence and sense of self-preservation. If they see that the fight has clearly gone south for them, it's reasonable for them to surrender or run away, eliding what would otherwise be the grueling remainder of the fight.
Like in the example above: instead of narrating that the level five characters mercilessly brutalize the goblin, maybe the goblin just immediately surrenders without putting up a fight—or hopefully the players didn't intend ill will towards the goblin in the first place!
There's direct precedent in some adventure modules for this as well. For example, in the Lost Mine of Phandelver adventure, the very first encounter in the module suggests this course of actions for the goblins the PCs encounter:
Four goblins are hiding in the woods, two on each side of the road. They wait until someone approaches the bodies and then attack.
This will likely be the first of many combat encounters in the adventure. Here are the steps you should follow to run it effectively:
—Goblin Ambush, Lost Mines of Phandelver, D&D 5th Edition Starter Set
Having said that...
I do need to slightly challenge the premise of this question. It's one thing if you're only expecting players to lose a few more hit points to a lucky hit by their enemies, but if you're expecting players to still use up spell slots, that implies pretty strongly that the outcome of the fight is very much not set in stone. Even at high levels, a spellcaster would probably prefer to use their cantrips if they don't expect to need their spell slots.
So while the above advice is, in my opinion, a pretty good rule to follow, it should be tempered with the advice that if the players can be expected to expend expensive resources as part of their effort to end the conflict, it's probably too early to call it.
Best Answer
Vehicle upgrades increase in quality as well as 'size' (for lack of a more encompassing term). For instance, your starship can get a Short-Range Scanner (range 5) that is Budget, Basic, or Advanced. Basic adds +2 and Advanced adds +4 to all related checks. Each range has a similar suite of options. Computers are a little more straightforward; they start at Basic (no bonus) then can be upgraded to Mk 1 (+1) and go up to Mk 8 (+8). Thrusters only provide 0, +1 or +2 bonus. You could of course get some of this early, but the higher your Tier, the more BP you have to use so you're more likely to meet the demand.
Does this offset the difficulty increase? Let's look at a difficult Computers check, assuming linear Mk increase over levels. I'm also assuming full skill ranks and that ability scores are allocated appropriately but not min/maxed.
Factoring in the Errata linked by OP, the ability to succeed against average DC's are pretty static across the levels, assuming averages.
Piloting's numbers would be similar, but shifted to be a bit more difficult, but Pilots also get bonuses from Ship size and maneuverability that would be hard to summarize into bullets. There is also the Ace Pilot theme to help do their job.
Ways to Increase your Vehicle Capability
1Insight bonuses do not stack, only the largest of these applies.
2Circumstance bonuses generally stack unless a GM indicates they don't.
For semantics, a Operative acting as your Science Officer, as above (Ranks [inc. Class Skill bonus] + Ability Mod + Equipment Bonus + Operative Bonus):
Lashunta's have the ability to add another 10% to each of these.
While it is now possible for any class to succeed at their role, it is still advisable to pick an appropriate Class, such as Operative or to have another way to gain significant Insight bonus.