An overbet on the flop

After you review the hand on HIT, here's the way the action went down:  The player in the Hijack seat was a loose-aggressive player who had routinely shown a tendency to overbet to protect weak hands. After calculating my odds and thinking over the prior action and putting him on a range of hands that included any pocket pair, any Ax and any two Broadway cards, I decided that even though I wasn't getting the right odds to call, since I had middle pair/top kicker and straight draw to go ahead and call anyway. The turn and river were both bricks that were checked down by both of us and Hijack turned over a King with a rag kicker to take down the pot. I really didn't feel that going all-in over the top of his bet would've folded him out due to his table image and, for that same reason, I was highly suspicious that the overbet was intended to protect a weak pair. Should I have folded the flop? Maybe floated and went all-in on the turn? My gut feeling was that he wasn't going away regardless of what I did since he had me covered and is a very loose-aggressive player. Tell me what you think.