I would push in. You say BB is solid, so if you just call, I think he has to know that you have a big pair and are obviously going to be putting it in on the flop. Even with implied odds, he isn't getting enough to try and flop a set, so I think the only play is to push; maybe he'll read you for AK and call with QQ. If he has KK, it doesn't much matter what you do.

You definitely should have shoved pre-flop as that flop can show you - the guy was an idiot so he payed you off even though an A and a Q flopped (worst two cards in the deck for K-K based on the pre-flop action).

Most good players will fold that flop with K-K when you shove (or fold to your min-reraise), that's why you shove it pre-flop and get him committed everytime; he's almost never folding that pre-flop.