Mini-Main Event $1K, KK Multiway

Blinds are 600/1.2K w 1.2K ante
60K starting
We have 80K
BTN has 65K
BB has 40K

BTN is a bit splash. We don't have a super good read. He has had some good hero calls, but I am thinking he might be a bit of a call station.
BB is newish to the table

Preflop: Hero has Kc Kh UTG
We raise to 3K, UTG+1 calls, MP calls, btn calls, and BB calls

Flop: Qs Jc 9s (pot 16.8K)
BB leads for 6.1K, hero calls?, 2 folds, BTN calls

Is anyone raising this flop? I just thought with 3 other players behind me, If I raise, I am going to be building a pot, and I am not sure where I am at.

Turn: 4s
BB checks, Hero?

What are we doing in this spot? Check/call the turn and check fold the river? Just hope it gets x around and that we are good? Bet/folding?
If I check in this spot, Btn can really punish us. I kept thinking that btn might have a flush and I might be already dead.