First of all, oop you should raise bigger than that. 12k min seems the way to go here.
Then I think we could go either way in such a spot. There is merit in checking. You control the pot size and you hide the value of your hand letting your opponent a chance to value bet a weaker holding and a chance to (semi-)bluff. By checking you dont let your self in a weird situation where you get raised on that T turn. J10s Q10s A10s are part of his preflop calling range ip as well as 66's but most of the time, you are still ahead on that T turn.
Betting has some merit too because you make the Flush and straight draws pay for the river. He might even have floated flop with a gs like AJ or AQ.
If he's very sticky and wont fold anything, you might want to check for the reason mentionned above, if you want to protect your hand from the draws on such a wet board just bet at least half pot.
Personnaly, i would check turn and reconsider river depending on his action.

Depends somewhat on our squeeze range and the amount of Tx we have , also on our flop bet range / strategy and how far up our range we are , on texture shifts we want to start blocking our range that continues to bet , mix between block and check for example AA is an easier bet due to unblocking Kx if we have some Tx we want to block also and mix in some checks ( like our worst Tx if we have them would be an ok strat so maybe a T8s Kts if we squeeze at some low freq rather than defend , The 3 elements of blocking is value / protection / bluff .... Well Ak isn't a bluff and doesn't need protection really , but can bet for value however we need some checks and im thinking this makes sense due to needing less protection than KQ and KJ , because at some freq we want some Kx that checks and a small amount of Tx as said , usually id choose my weaker Kx for obvious reasons we have less hands that calls us however here AK needs a lot of less protection , knowing we need to mix block and check id imagine pio would go something like close to 50/50 , checking AK here seems the play.

One final point your sizing is to small preflop and you've left yourself open to a multiway scenario , as an exploit also when you stretch the pot size , check calling becomes super easy as villains will overbluff due to pot size and chips committed and we can just go into c/call mode and never fold this combo.

I'd check because of your flop lead. Button has way more Tx than you because of the flop lead. Let him bluff if he wants and control the size of the pot if he doesn't. Agree with everything Swiss said.