Get the engineering degree
Forget about whether you want to sit behind a desk or an instrument panel for this decision. Neither the military or any airline cares what your degree is. The non-flying job market does!
If, at some point in the future, you decide flying isn't for you or you can't pass your physical due to any number of reasons, you'll have a GREAT fallback career option. While I won't say an aviation management degree is worthless, the engineering degree will open a lot more doors when you need a paying job...
Get the engineering degree. Period.