I'm trying to kill two birds with one stone with this post - validate that I understand combo-counting correctly, and make a flop decision when I'm playing with a short stack and I flop overcards on a paired board.

Let's say I am up against a villain who I believe will limp-call with this range:
any ace
any connected cards 54 or higher
any one-gappers 53 or higher
any pocket pair
any 2 broadways
any 2 suited cards

A2 A3 A4 A5 A6 A7 A8 A9 AT AJ AQ AK - 12*16 = 192 combos
54 65 76 87 98 T9 = 96 combos
53 64 75 86 97 T8 J9 = 112 combos
22 33 44 55 66 77 88 99 TT JJ QQ KK AA = 13*6 = 78 combos
KQ KJ KT QJ QT JT = 48 combos
K2s-K9s = 32 combos
Q2s-Q9s = 32 combos
J2s-J8s (already counted J9 and JT) = 28 combos
T2s-T7s (T8 and T9 already counted) = 24 combos
92s-96s = 20 combos
82s-85s = 16 combos
72s-74s = 12 combos
62s-63s = 8 combos
52s = 4 combos
42s-43s = 8 combos

710 total combos. Correct so far? I hope so.

Now let's say I have 30bb in my stack, he has me covered, he limps, I raise to 7bb with AKo, the blinds fold and the limper calls.

Flop J44r, let's say the pot is 14bb to account for rake, and I have 23bb behind.

Let's say this is the subset I think he would actually call a bet with:

A4 AJ AQ AK - there are 2 4s, 3 Js, 4 Qs, 3 Ks and 3 As unaccounted for, so 6 + 9 + 12 + 9 = 36 combos
54 = 8 combos (only 2 4s are left)
64 J9 = 20 combos
44 JJ = 4 combos (is that right? There's only one combo of 44 left, and 3 of JJ?)
KQ KJ = 21 combos
K4s = 2 combos
Q4s = 2 combos
J4s = 2 combos
T4s = 2 combos
94s = 2 combos
84s = 2 combos
74s = 2 combos
42s-43s = 4 combos

107 total combos, and I'm only ahead of AQ and KQ which are only 21 of those 107 combos. Correct so far?

Now on to the flop decision. If this villain donks into me on the flop for anything over 8bb this feels like an easy fold.

If the villain checks to me:

I don't like shoving because we're behind about 80% of the time we're called.

But I don't like checking behind because approximately 85% of the time he has unpaired undercards that have 6 outs to beat our ace high and I don't want to give him a free card.

A bet of even 7bb would make it unprofitable for him to chase his draw, but I don't like it. With AA or KK I shove there 100% of the time and expect to be called by worse, so why turn my hand face up? And then I have a half pot behind on the turn and am going to have to decide between putting the rest of it in anyway or folding after having put half of my stack in the pot. I don't like that option either.

I honestly don't know what the correct decision is here and will appreciate insight.

I think both jam and small bet are viable here. We aren't ahead when called very often, but we can be and we still have 6 outs against almost all of the calling range. Plus, when he folds we deny a lot of his equity.
It's not just about being ahead/behind when called on the flop. Bet small also accomplishes a lot and you are deep enough to play turns, but it can lead to some awkward spots as well like if we face jams (which we might have to bet/call??) or when v calls and turn bricks, which is imo a fine spot but can be tough to approach.

It seems like you understand combo counting but I don't think you should usually try to directly count combos while playing. Having a loose idea of relative strengths and hand regions is typically enough.