It's absolutely true that villain could have a fair amount of 6x hands in his range {A6, 76, 65, 86, 64}. Although it's not correct to call a 3bet with those hands in this spot, they could very well do it. And since it's a mistake for villain to make these calls, you are already profiting by your preflop decision, because as we know the majority of the time your hand will hold up against theirs.

On the flop, I wouldn't be cbetting AK here, which against this bad of a player is like the only hand I wouldn't cbet. I'd obviously cbet all of my overpairs against this player (and probably against most). But against this player who likely calls a 3bet with all pps, and a ton of broadways, cbetting is likely incorrect with AK here. He isn't going to fold a pair, and the broadway hands that he does fold, we beat. So check it back, and pick off bluffs on the turn and river.

As far as the turn play. AK is at the VERY top of your range. I doubt you ever have a better hand in your range here. To add to that, villain is known to play badly. Of course it would be pretty bad to c/r all in with this much behind with KJ/KQ/KT/QQ/4x/Ax, etc, but villain doesn't know that, and could very well make that play. 6x is not the only hand he gets it in with here.

If this was a reg or someone respectable, then it would really look like AA, and you could maybe consider folding, but even then it would be close.