"i dont put him on A10 or even 88 because calling a 3 bet with effective stack sizes under 40bb isnt too good" this doesn't mean that HE is not going to make the call preflop, remember you`re in a 4$ tournament where you assume all the players are bad.
There`s also no reason to slow down on ther turn and give him a free card in case he`s drawing, i`d probably bet somewhere around 7.800-8000. If you get check-raised you`re def in a tough spot and you might even think of folding as bad players are not that inclined to making semi-bluffs check-raising on draws.
If you decided to check i`m more inclined to a check-shove rather than a check-call as now there are straight and flush draws out there and you want to get as much value as you can out of drawing hands. You will also get value from high pairs in case he decided to just call preflop.