HOTD: Flopped Broadway in three way pot

I have AJxx in the BB and four of us see the flop of KQT rainbow. There's .20 in the pot and SB leads for .10. I min to .20. Next to act raises to .35. Fourth player folds. SB calls. I flat call. Turn is a blank. Checks to me and I pot it for basically my entire stack. Player behind me over-pots. SB folds and we chop it. In retrospect I think the correct move would have been to put in min-re-raise on the flop to try to get another bet out of the SB who I am guessing had two pair or a set, maybe with an open ender for a chop if an Ace comes. I'm pretty sure that 3rd to act has broadway as well. I don't have any redraw at all so part of why I called was to wait for a safe turn but I'm not sure if that was the most EV play. I'm pretty sure I'm chopping with the player behind so the question is how to get the most money from the player drawing. Given how safe the flop is I think I could have gambled more as opposed to a flush drawish board.