I got to the river with only about 15BB's behind, I think I should have checked behind on a river like that as I can't get much value from anything. I think I only get called by AT, QT, JT, T9 that have a spade. I think I went too thin on this river. How should I have played this hand?

At micro stakes, you want to be going for thin value because the players are weaker and you get called down a lot lighter.

It's perfectly fine and good to go for three streets here, especially with the Ks in our hand which blocks combos of flushes they could have. A lot of players at these stakes won't be able to fold their Tx. Some may even pay you off with 77-99.

Might want to size a bit bigger on the turn, ~5200 or so.

I don't know what you mean by "too thin" on the river...you were all in, it was for over 1/2 pot, and there's no way you could possibly make the bet less thin when you have all of your chips in.
I don't really see any problems with how you played this hand, except I probably make the sizing bigger on the turn vs. SB in this situation. He's pretty deep relative to the table and you gave him 5-1 on the turn to hit one of the many possibly draws out there. It's very possible he called you with a hand from the SB for a min-raise that connects very well with that flop/turn. I think I like the small turn bet better if you have AA with the A of Spades. Literally any 2 cards are in his range here. He probably tells you he has a T at some point if he does. I don't think I bet this river as his action so far makes it very possible he has a random 4 in his range and tripped up on the river. To me it looks like you dodged a LOT of outs on the river and gave him a great price. So ya, I see this as a pretty scary situation for KK but if you were willing to go with it (which it seems like you were) only thing I'd change is the turn bet sizing to something 5k-5500 to stick with your sizing from the flop bet and give him a little worse odds/increase your value (I'd probably go about 7k to try to make a drawing hand make a bad math call). If you go 5500 on the turn he's getting like 3-1 on the river call and may think you're running a crazy bluff and call you down with a hand like As6x or Ts7x (I don't think he had a T though).

When you increase your stack by over half without a showdown, it's always a good thing.

more turn to set up river jam

fine to go for thin v as our hand could look like missed f draw so getting looked up lighter

In these spots I typically go for the river check to induce a bet. It's amazing how often you can entice a villain to bet the river with a marginal hand or missed draw here (he could have the As). I might miss some value by not going for three streets but I think inducing the bets more than makes up for it.

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we last to act for some reason tho doesnt say sb checks throughout the hand

id probably play this as a two street hand tbh. bet like 4k otf and shove ott.

Ya I just noticed that, this came straight out of PT4, not sure why it like that.