The hardest part about what you are trying to accomplish is the 4 year degree....because if you stop to concentrate on "my career" it becomes exponentially more difficult to complete after you have gotten started in said career and life intrudes.
Personally I would knock out ANY 4 year degree while flight training as much as you can outside of the university....because noone cares how much you "paid" for your training or if you flew a cirrus or a c-152.
But I would train at a school that has a multi so you will be able to get hired there and then teach in the multi to build the time.