Hi,
I'm at Embry Riddle, Daytona Beach's campus; and I'm doing exactly what you are saying: I'm an Aerospace Engineering major, but I'm passing also my flight ratings (finishing my CFI right now).
Now I don't know how it works in other universities but fitting my rating in my bachelor degree wasn't a problem untill now (just became a sophomore).
I know a lot of things are said about ERAU, it's expensive, and it's probably not the funniest flying you'll do, but in the end you have a degree and your ratings from one of the best universities (especially in AE).
I totally agree that you should have a back up plan. Most of the people I see flying at Riddle are Aeronautical Science majors, which I don't really understand the point, cause what can you do with that other than being a pilot?
If you do your ratings but a different major (like AE or any other), you are as qualified as an AS major but you have a backup in case.