Super meta check/snap river

I had Aa78ds

Heads-up

I raise, opponent three bets

I forget if I called or four bet. Think I 4bet or clicked it

Call

Flop aa3

I bet. He min raises. I min raise. This goes back and forth until 70% of effective stacks are on. I did the last click back. I bet ~20%. He calls

Now I chk snap

My logic; he could be floating a non-zero percent, and if he has quads I'm dead, he's probably (probably- this is where it gets weird);

Do we make more money shoving versus villain? Or does c/c work just as well?

Also; he had ak3*

I chk snapped

He should of b/f half

Or b/c and induce 1 in a trillion "GTO" node lock logic