Bottom pair + FD facing a flop 3 bet / Should I 4 bet shove?

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Bottom pair + FD facing a flop 3 bet / Should I 4 bet shove?

200NL Villain is unknown.
UTG $5
MP & co fold
Hero BTN 8s7s call
BB calls

Flop ($16) Js 7d As
UTG ($covers) cbet $8
Hero ($200) raises to $23
BB folds
UTG raises to $56
Hero ?

This 3 bet didn't make a lot of sense to me because it doesn't set up a turn shove. The pot will be $128 and stacks will be roughly $144. If Villain does have AA, AJ, JJ, would he not size larger oop? It felt like a "feeler bet" where he would 3bet fold a hand like Ak because he is worried about 2 pair or a set being 3 ways to the flop. I went with the shove based on this logic, blocking 77 and A7s, then decent equity when called I think.

Side note you can run it multiple times, I am not sure if that plays into it at all because of dead money and ability to chop the pot on multiple run outs. I don't want to get blown off my hand on about 70% of the turns I don't improve on or a turn Like AJ79, where I pick up additional outs.

In villain's shoes, what is correct 3 bet sizing? In my shoes better to call the 3 bet or shove?

Snowie advice
Preflop snowie has 87s here as a 3bet as 0EV and calling -0.22EV. I usually prefer to flat btn vs utg and have a more snug 3 bet range.

Flop - Shows EV of calling to be higher then any raise size. I think it's because we don't have AA in our range and only some JJ combos. Also don't have A7s in our range. I guess my flop raise is not a thing despite having a lot of equity.

Against a 3 bet snowie has 4 bet shove -2.57EV and calling +6.91 EV.

What is your take on this hand?

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