Getting counterfeited--how to stay strong?

[The Ram]

Stars 2-4 NL:
In SB w/ JT of hearts. EP min raises to $8 and 2 guys call behind him, I call and so does BB.
Flop: AhJs3h.

Pretty solid start for me, so I plan to checkraise the EP raiser. I check and BB check, but EP checks too. 1st coldcaller bets min ($4) and button calls, and I pop it up to $16. Folds to bettor, who min raises me the $12. Button folds and we're heads up.
Now, I know the bettor from several previous games. He's kind of a maniac, or at the VERY least completely unpredictable in his hand strength, so I don't know where his reraise puts me and I call. (Mistake?)
Turn: 10s (AhJs3h).
Hmm, so 2 pair now to accompany my flush draw--I decide to go for another checkraise. Opponent bets $56, I make it $168, and he flat calls.
Based on my experience with this guy, this puts him on a big ace. AK or AQ. I could be wrong, but he has played top pair exactly like that a few times already and he'd have his gutshot, so I'm fairly confident it's at least some kind of ace.
River: ACE!!!

So, my 2 pair are no longer worth anything and my flush didn't come in. However, if I just check to him and my read was WRONG about the ace (maybe he just had a flush draw--he's crazy enough to play it like that) I'm sure he'll shove in and I'll have to fold.

So, what is the play here? Or in any case where your 2 pair is counterfeited but after being the aggressor for the whole hand, you will hand it away if you check.

I bet $36 (he had about $110 left)--figured it was just enough that he wouldn't just shove all in sensing weakness.

Thoughts on this (and related situations)???