I think your line is fine but I shove river.

The guy's remaining stack is 1/3 pot, we beat some piece of his range, it's impossible to give up the pot here until you 've some strong reads. And i prefer shove too until i've good reason to think that x/c or x/f is better.

if he's LAG then won't he have a much wider bluffing range here than calling range?

You should size it that you have roughly 60%-120% left oTR to shove on or bet turn to shove river, imo. I think you can call OTR, villain reps a narrow range and folding this will make his bluffs super effective for that price.

What if he's always bluffing in this spot ?

I don't like ur size as well. As played checking river is def better then betting, V def has some weak flushdraws in his range that would mandatory bluff river.
For that reason and with that spr I can not see us folding here ever.
Xc is a play.

Yup, as played call river, but seriously work on your sizings. This is a serious professional mistake here : just see how you build the pot on every street... clearly not optimal and not well planned. Also you need to consider your perceived range: in this situation, you look very face up with your sizing, and also very strong, which I think is not great here (clearly you'll get a ton of mucks on the turn from hands you beat... some decent flush draws might stay there. But seriously, I think many players should (and will!) fold AQ or worse on this turn versus your line.

Anyway, I would first and foremost make it bigger preflop, as your 3b range from BB is surely strong anyway. Postflop, I do see merits in betting very big on the flop, because many hands will call now and fold later, and also it allows you to shove turn, making it a 2-street game (on this board texture, I kinda like this option). You can also 3 barrel, but please make it smaller flop and turn to get a better stack to pot ratio river...