This is a tough one, and the MAPSS program can be great in some respects, but it isn't a guarantee that you will get in elsewhere afterward. Prestige warrants prestige and you get some great work and LOR and acceptances coming out. But I have a friend who did MAPSS and still got rejected everywhere again, and had to reapply yet one more time. So it's not 100%
If you have a funded offer to go elsewhere, I consider that. If you really hate that program, imagine it as a fully funded MA, giving you two or three years to flesh out your project and work on yourself. It's not duplicitous, it's responsible.
Worth considering - Chicago's program is expensive, and you won't be applying for PhD programs again this Fall, you're going to be taking classes for a year, then applying that Fall, so you're going to have to support yourself for a year in between not in classes. Think about what you want to plan for that year as well.