The button is not a very aggressive player preflop or postflop, judging by the stats. With 3 people that already limped in front of the button preflop, I dont think the Button would raise AJo,KQo,or maybe not even TT, and he would also limp in with J9s. On the flop, the button can have any of these hands. On the turn I dont think the button would 3 bet with KQ here so that hand is ruled out. He may not even 3 bet with TT, and he may raise TT preflop, so this hand is also highly suspect. So it looks to me that the button either has AJo or J9s and possibly TT. If you assume this hand range then you are a 4-1 underdog on the turn to have the best hand, which means all you really have here is a crying call on the turn and river IMO. So no I would not cap the turn and no I would not lead the river, not against this opponent.

Also, even if you did assume the button would 3 bet the turn with KQo, you are still a 4-3 underdog which means you should still just call down. I also did not include hands like KTs or QTs in the button's range becuz I am almost certain the button would never 3 bet this turn with these holdings.