Buy the hat yourself so you can get the lipstick so you can then get your hat back.
In fact, that's the whole event in a nutshell. Every item one side wants is a reward for trading items to the other side... yup..
You'd think after so long in the temple, they would have learned to talk to eachother. lol
But anyways, yea, you need to have already brought some of the requested items yourself to get the ball rolling.