It doesn't matter what your degree is in as far as the Major's is concerned. Most state within the requirements that a 4yr degree is required. A few will use the term "preferred". When it's termed like that, rest assured it's a silent requirement. As far as the Regionals and LCC's, I'm not sure. Those who are flying can attest to that more so than I can.
With respect to flight training, I would look at all aspects of training program and process; whether Part 61 or 141. That would include costs, flight instructors and the quality of instruction, placement successes...if applicable, maintenance history of a/c, type & condition of a/c, and location etc., etc.
I wouldn't worry too much about time constraints. Seems like all those who have recently started their training along with those of us who hope to starting soon will have ALL the time we need in order to amass the required minimums. At last glance, the norm seems to be 1200-1500tt and 300 multi.
All the best.
atp