[M.B.E.]

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As far as what the SB had, after I folded to his push, he showed an unimproved AQ to take it down over UTG+1's unimproved AJ. Putting aside the fact that his push was pretty stupid with no side pot and a monetary advantage to knocking the player out (correct me if my thinking is flawed here), I just wanted to know how I could have avoided losing half the pot to him.

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The SB's turn bet with AQ was fine. He figures, correctly, that you have no pair because you checked the flop. If you have AK or AQ you will probably fold to his bet. Even though it's a dry side pot, there's some chance that AQ currently has the allin player beat, and if not then it likely has 6 outs. (In addition, the board was paired, so if the allin player happens to have AK, then the SB has six outs to tie plus three outs to win.) Finally, at this stage of the tourney the "advantage to knocking the player out" is not a real consideration.