While we're on the subject, I had a similar question about loot drops. So let's say you move to normal tile as your first action of the turn and roll an encounter and a treasure. If you defeat that monster at any point, do you auto pick up and get a free equip action for that treasure?
There's no such thing as auto-pick up when defeating a monster, you always have to spend a Search action to get the treasure on your tile. Same thing with the trap, the treasure stays in the room (regardless of the trap disarm roll) until someone picks it up with a Search action.
If you kill the monster with your second action in a turn, you can reasonably expect one of your teammates to be really nice and polite, and pick up the treasure for you!