I think I'd rather bet the flop. Decently scary board for our opponent, so I don't think you will get raised very often and you can easily get folds from hands like Axxx or Txxx that you would love to see not in the pot. And you have an easy fold if you get raised. I'm not sure I would three bet this hand either, although I certainly don't think that three betting is an obvious mistake and it may in fact have a higher return than calling. Obviously we can expect the button to raise pretty wide so I am on board with using some decent hands to punish the raise.

I don't see much merit to betting the river. Our hand is to good to bluff with; if our opponent beats ace jack he will usually call on the river. Our hand is too weak to value bet, it is unlikely that he will call with like QQ or something.

I like the 3b pre, on the flop we have a big range advantage hence we can bet. Big size makes sense as we want to charge the FDs , i ld consider a smaller size or even a check with a much stronger hand ie a hand i ld be willing to slowplay AAKsJs.
On the turn i like your reasoning , although i doubt many FDs fold for that price.
On the river i see merits for both x-c , x-f and betting.
Our hand is probably weak to value bet - true, so if we are to bet we mainly will do it as a bluff.
Our hand is too good to bluff with ? maybe/ maybe not depending on the range we end up getting here with.
Our hand is a decent bluffcatcher? Bluffcatching with AJ is a bit too thin for my liking. Will opponent bluff this river often enough for AJ to be a profitable bluffcatcher? I doubt this is a spot where opponent can show up overbluffing , on top of that we ll be having a lot better bluffcatchers here even if our flop checking range is not balanced.
So as our opponent probably will not have many bluffs on the river i believe we can make an exploitative over-fold here.
So x-f or bet-fold is the highest EV line here?
Its probably opponent dependent but with no reads and with the 2c in our hand i think i 'ld opt to go for a bet.

Feels like this an awkward spot to 3b hands like this, mainly because it could end up with the spot like this on the river. Flatting raise on BB makes a lot of sense to me.