Thin Value on the River?

Was hoping to get some feedback/ insight on this hand.

Pre flop : I think is okay? I lean towards playing a 3b/ fold strategy pre flop (although, I have seen a lot of pop playing flatting ranges on BTN). When UTG 4b I think flatting is fine. Obv, we are not 5b & I think folding is out of the question - especially considering how deep we are (138 bb's to start the hand).

Flop : Villain bets 25% & I think our hand functions as a pure call. Nothing too crazy. Straight forward.

Turn : ck/ck

River : This is where I was hoping to get some feedback. Villain checks to us & now - do we bet or take our showdown & ck back? When trying to find hands that we beat that we can get called by - maybe 99/TT (although idk how often we are going to realistically get called by those hands), versus hands that beat us (AA, KQs, QQ, JJ). When villain checks river after the turn goes ck/ck I think his range is capped - 99/TT, QQ, AA. My line of thinking was - if we bet river & villain has 99/TT he is going to fold & if he has QQ, AA, KQs he is going to find a call so felt like our hand was too thin to value bet. Is this reasoning faulty? Should we be betting this river & if so, what sizing should we use?

Thx in advance to any & all responses.