Fill me in on something here. Is there even such a thing as a degree in Aircraft Maintenance Engineering? I say no.
A degree in Engineering does nothing for an AME. Much like a diploma in any aircraft maintenance course does nothing for an engineer. (P. Eng)
As "Krystal and Chips" mentioned, it never hurts to keep your options open. But who can afford to go to school for that long with no income? I say stick to one thing and evolve as you see fit.