I lived 15 minutes from the ASU Tempe campus (where Barrett is located) for over ten years. It’s not well respected as a school here, I’m just speaking from a local perspective so it might have a better image elsewhere. It’s not that there’s anything wrong with ASU as a school, but paying extra for Barrett seems rather silly. Also, and this is just personal conjecture, not many people out of state will even know about Barrett, so ASU as a whole is what their preconceptions are going to be drawn from.
You’d be surprised, I found that private liberal arts colleges generally have very good financial aid. It would cost the same for me to go to ASU as it does for me to go to Cornell University.
I would suggest you apply to a number of reach schools and see where you get accepted, before going to ASU. But choose what’s best for you, and if that’s ASU, all my best wishes to you.