Don't know how much BvB changes this... At $10nl where I used to see this limp/3b alot more in EP, it was almost always AA/KK. Raise the flop may push out AK/AQ/KQ type hands. Keep in mind if you do, you'll have >50% of your stack in the middle so think carefully what you'll do to a shove and subsequent streets he just calls.

As played, with his AK hitting the Turn, I think it becomes more about pot control. He possibly slows down on the Ri worried you're sitting on a 67s, 78s, 89s,55,44, A8s kinda hand?... or his repping the K w/ JJ-TT didn't get the love he'd hoped. Either way, I'm checking this through. If it's a very aggressively played busted FD, I think he'll fold. If it is JJ-TT then he might call something small. But if it's anything Kx (w/ AK fitting nicely into that... you know, we limp 'cause we're gonna get tricky when we spike our A or K on the flop, but then get raised, so sh!t, my hand is premium so I will 3bet!) or, god forbid, AA/KK, you're f-ked and would probably have to pay off a shove.