A solid reg is not x/r one pair hands in this hand, that is the point. And he is not checking river to x/f a value hand.
When you bet here with such a huge sdv hand it does indeed demonstrate that you are simply trying to win hands and not actually think about his line/range vs our line/range and that we beat just about everything except for the stuff he is never folding.
The 4xhh is the one hand that I can imagine we might be able to fold out, that is currently beating us. As you yourself pointed out, he likely only has one combo, 5
4
, and that still may have bluffed river.
Putting villain on TT here is just bizarre and wishful thinking. Thinking he'll fold it is even more wishful thinking.
As for a value range otr, I actually probably x/back AKhh. There is no real value to be had from his range, unless he is indescribably bad.