[Jesuitical]

He could definitely play AA and KK this way, intending to wait until the turn to check-raise (hoping you catch something that's not a set or commit yourself to the pot with a second bet) rather than bet out on a raggy flop. If you decide to give him credit here for 3-betting a reasonable range, his range is AA, KK and AK, which still makes a push bad. Of course if you assume he sucks his range is much wider.

One of the points to letting him outdraw you is that you're not playing for your stack with a weak overpair knowing nothing about Villain. Another is that you can re-evaluate on the river. If he never bluffs the river, there's a good chance the pot is checked down, and if he bets the river big you should fold. He probably bluffs the river some of the time because of how weak your line is, so you can call a smaller bet. I think checking behind is +ev and more suited to how you should be playing extremely early in a tournament.

Edit: I did a HORRIBLE job of articulating this the first time I tried -- to anyone who read that version, I'm sorry you had to go through that.