MP - $37.50 - 50/25 over 10 hands
Btn - $71.22 (Hero) - Kd Ks 5d 8h

Preflop:
MP raises to $1.75, Btn raises to $6.00, MP calls $4.25

Flop ($12.00) Jh 5c 2h
MP checks, Btn bets $8.00, MP raises all-in for $31.50, Btn?

Preflop I had a limited sample vs this player, but since I don't have super good KKxx, I thought 3betting and pushing my equity advantage preflop and isolating this player would be a good idea. My main question here though, is what kind of strategy do you like in this spot? From a pseudo-optimal approach, maybe betting $7 would be good so we set up stacks for a $24 turn shove into $28 on good turns with our range? From an exploitative approach with our exact hand, would it be a good strategy to bet half pot for $6, and fold to the jam? Or bet bigger like $10-$12 and not fold to the jam. I'm not really too confident in this spot on my bet/stacking off ranges. Some bare Jxxx hands have 42% equity vs us. Theoretically if we had AQJ9ds without a backdoor FD at this SPR, we would need 42% equity to stack off on the flop, but wouldn't have it because our opponent could have us dominated. That means hands like that should fold and betting KKxx here can deny a huge amount of equity from those hands. But Jxxx with a FD has roughly 65% equity.

I guess thinking about it critically, blank Jxxx without a FD should probably vs population betting 50% pot, since population is not showing up with a ton of bluffs after 3betting, calling ranges can we wide from recreational players, so maybe betting 50% here sounds like a good option, because you can fold easier to a c/r without putting yourself in a marginal spot to need to call it off, but you may be more likely to get value from QQxx as well and Jxxx too. What are you guys' thoughts on this spot? Would your answer change at all with the same hand, but Ad As 5d 8h?