my two cents:
Betting ourselves
The tough thing with this spot for me is that on the one hand it's hard to expect villain to call down three streets unless:
1) he a strong hand anyway and is never folding
2) is very passive
3) doesn't give us much credit, due to table dynamic and our image / history etc ( we don't have this info available, which I feel would be crucial in decision making OTR)
Check raising:
It's also tough to expect villain to bluff when we check to him OTR especially since we have show strength preflop, OTF, OTT already, unless:
1) villain is very aggressive
2) we have already seen him perform this kind of double float
3) he thinks our range for stealing and barrelling is very wide and weak and is exploiting this by floating two streets and betting on the end since he thinks we are almost always x/folding here or something.
4) there is a very agressive dynamic between hero and villain.
So if we don't expect villain to bluff in this spot we are only getting value from hands that villain would never fold OTR anyway ( Kx, straights, weaker FH's.... maybe Jx). This would mean that our reasoing for x/raising vs betting ourselves is that one would yield a greater profit in the long run that the other.
Without further stats or reads (which I think are crucial in this spot) I would prefer x/raising on the bigger side just because I find it hard to believe that villain would call down with JX or worse. Therefore I would prefer to give him the opportunity to bluff and also win a bigger pot if he has a very strong hand like QT, KX and weaker FH's that I think he will at least bet/call with.
Re: sizing OTR I would make it $152 and give villain 2:1 on a call ( I think) and go broke to further aggression.