River donkbet on scare card

[CrashPat]

Live 3-12 spread limit game at 3am or so, I think we are 8 handed, maybe 9. I play this game as 12/12 with 2/3 blinds, I have found that in small spread limit games it works very well to only ever bet the max, except very rarely to squeeze out half a bet of value on the river in a tiny pot.

Villian in this hand is BB and seems to be an ok player. He plays too loose, and seems over aggressive at times. I remember playing with him on one other occasion and he was not a complete donkey their either, I have only been at this table for about 30 minutes. If anything he likes to be tricky postflop.

Hero has 8[img]/images/graemlins/heart.gif[/img] 8[img]/images/graemlins/diamond.gif[/img] in middle position. All bets and raises are 12 dollars in this hand.

1 limper to me, Hero raises, folds to villian who calls.

Flop (2.75BB) 8[img]/images/graemlins/club.gif[/img] K[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/diamond.gif[/img]
BB checks, Hero bets, BB raises, Hero 3-bets, BB thinks for 10 seconds and calls.

Turn (8.75BB) T[img]/images/graemlins/heart.gif[/img]
BB checks, Hero bets, BB calls after thinking for another 20 seconds

River (10.75BB) K[img]/images/graemlins/club.gif[/img]
BB bets, Hero ?

His hand range is very large here I think, as he likes to try to be tricky postflop, he might call preflop with any 2 broadway cards, any suited ace, and any pocket pair. I am not worried that he has TT, JJ, or KK because he would have 3 bet preflop with any of those holdings. I am very worried about KJ, KT, but I think he could do this with KQ, AK, or even a lesser hand if he thinks I will fold.

I am obviously not folding here, and my immediate thought is to call, but what about raising? I'm not sure if he has KQ or AK enough that he would make a raise worthwhile. I think I am calling a 3 bet and killing myself when he turns over KJ.

edit:
wow I am retarded, had that I was worried about AQ.. hmm, let me see that hand chart, does a straight beat a full house?