I like betting the flop, but I'd go for a bigger size, maybe $75 or $80. On such a draw heavy board, you
want to size it such that draws don't have the implied odds to call against you.
Once you are check-raised, I think that your beautiful overpair actually becomes a bluff catcher. Is Villain really going to check-raise AJ on such a scary board? Probably not. Especially if Villain is limpy.
The question then becomes whether you should continue. You have 3:1 pot odds which would imply that you need 25% equity to continue. With 6 outs, you have about that much according to the rule of 4&2, so it looks like a call.
However, this isn't an end of action spot. There are more rounds of betting to come. So if Villain bets pot on the turn, you're not calling $100 to win $307, you're calling $407 to win $707, which means that you need more like 36% equity. And that doesn't even factor in the river. So if Villain is on a strictly value range, I think that this is definitely a fold. And even if you throw a few bluffs in, it's still a fold, but most people wouldn't look to bluff on such a scary board.