So basically by the turn we have his likely holdings as AA/KK, QQ/JJ, AK, AQ, in that order? (I'm not sure I read your description correctly, should QQ and AQ still be in there after his turn bet? No matter what I think my following question is valid.)

My question, then, is "why do we want to get to showdown against this range"? Out of the hands in his range we beat only the two least likely, and QQ if we include it. If we are ahead we aren't getting paid because we never raise (and even if we did raise he'd find a fold with all hands except maybe AK). I'm not doing the maths but showing down here sounds like a losing proposition to me.

I believe a flop raise or a turn fold is in order.