Generally....
You will have no problems getting hired at a regional airline without a degree. Once hired, upgrade to CA is based only on seniority, and being able to pass the training (most do). A degree would not be relevant.
But compared to the best majors, a career regional pilot will work twice as much for about 1/3 of the pay. That's not an exaggeration at all (I've done both).
It's very rare for a pilot to get hired off the street by one of the top majors without a degree. The ONLY reliable path to that (in the US) is to take a job with a regional owned by American Airlines... they will have a flow program which allows senior regional pilots to transfer to mainline after a certain amount of years and experience (also requires job openings at AA). This will typically take longer than applying to majors off the street, but it looks to be fairly reliable and predictable for the next 10-15 years due to retirements at AA. Bear in mind that flow programs can change, and have failed dramatically in the past.
Also worth noting that the pilot group culture at most majors is pretty white-collar, ie wardroom not goat-locker.
In your shoes (GI Bill) I'd probably do a university flight program with R-ATP mins.
Worth noting that it would be possible to get an online degree while working at a regional if you're dedicated, although if you have a family that would be rough for a few years.