id have probably 4bet preflop - i wouldnt be scared to get this in preflop but it would not be standard - with JJ it probably would be against this player.
As played hes reading your hand pretty well so no matter what you do it sucks - he can easily have two cards or be value betting you to death

i probably do something on the turn - he has to act on the river first so he wont bet AK if you have shown interest on the turn, but he will vbet everything youre behind. If he check raises the turn then its an obvious fold