ooo nasty hand to call with at this stack size imho with little info to go by.I"m guessing there are antes Stu so for me i"d mostly fold here and if i"m feeling jiggy push the whole lot in pre and put the bra on my head.
As played - Forgive if this comes out in ramble ,he checks and if you bet here, the best case scenario is that villain will correctly fold their draw and you win obv , villain may/could make a mistake and call a draw or call a pair KK QQ JJ TT 99 etc or have a better ace so a turn push can fold out a lot of worse hands but obv u get called by better mostly here [if he calls].
When you describe his actions on the flop i cant help but think his act of weak means strong,you"d have a better feel for it with it being live..So i"d probably check the turn here,we give up some value by giving his possible flush draw a free look at the river but if no heart comes we get to see his action on the river..If no heart [or even if a heart comes] and he checks again i"d check also and get to showdown , if he fires out on the river no matter what comes i"ll spew up and fold everytime unless its a 5 ..