I cap the flop. I think your equity is good with UTG and Button along for the ride. It also gives you some post flop control and info.

I like a raise on the Flop. Protect against A or K overcard holdings. You also gain some info.

Honestly I am not sure about the turn, but some thoughts I have are:

1. You won't be able to protect against a flush draw on the turn, and there are not many gut shots on this board.

2. BB will not often lead the turn again with 3 callers if he is not holding a pocket pair, so I think AK is less likely. He may have an AKs flush draw.

3. You may not be able to fold if BB 3-bets.

4. You still need to protect against A and K holdings.

5. A turn raise may buy you a cheap showdown.

I would probably raise to buy some safe river cards, charge the flush draws, and maybe see a cheap showdown. I might call a 3-bet. I might check the river through if given the chance.