Hi @Quinine welcome to this part of the forum.

What was your reasoning for the 4bet after the BB called? I assume it was to fold and get HU with the dead money.

I think you can only check the flop and turn with your gutshot. 99-QQ is well within their range which aren't folding to a bet, AK and draws are prob calling too.

Betting sucks because of the size of the pot, even if we bet 40% pot it's still a large chuck of our stack. I just check back the turn pretty much giving up on the hand.

What do you know about BB?

When a reg is just calling here, usually he will have a very strong hand, AA, KK and he wants to keep you in the pot as well.
With AK, probably he will make a 4bet to isolate himself with the short stack player.

I 4bet because I didn't want him in the pot and I wanted iso his passive call. maybe -EV but I think in this way

I don't think I'm a fan of the 4bet preflop. It's not going to be terrible to iso the SB money but we cannot rule out the possibility of the BB trapping with a big hand here also.

Post flop then if we are going to bet I feel we need to do it on the flop. I'm not sure how much value there is in doing so though, the BB should not have many underpairs to the board when he has called your 4bet pre. We can't rule it out obviously but I would expect the bulk of his range to consist of unpaired high cards. This board will have connected pretty well with these types of hands and we have some equity with our gutshot so I think I prefer checking this down.

Also it is worth noting that even if we do successfully bluff out the BB we will still likely have to improve vs the SB to scoop the pot so my preference here is to try to realise our equity as cheaply as possible as our bluffs need to work far more frequently to turn a profit given the side pot dynamics.

I prefer just a call pre being in position, but if you have info on BB that he will fold often then 4 betting can't be bad, but we do have a decent hand that doesn't need to bluff.

As played I can't see anything to do but check and give up on flop and turn. Villain's range for calling a 4 bet just connects too much on this board texture

I'm not a huge fan of the 4bet pre. When bb overcalls 9bb I'd expect him to be on the stronger side. If I was going to 4bet to get this HU I'd like a bigger size but I think calling should be better.

Postflop assuming bb is pretty strong preflop, which he should be after calling your 4bet, he shouldn't be folding this board often. The only thing you get to fold with a single bet is 99- and I think firing multiple barrels here vs strong range won't be stellar.