Can my commander force me to shave my mustache because he doesn't like it, even if it fits the regs?

No in regs, but I suggest you do it as a dodge, and wait for that "trucker" to hang himself/herself screw up on regs with others or you or him/her move on. Watch this "trucker" they don't like you, or your mustache may be non-reg by a hair in their eyes, Dodge this simple problem.

It may not be within his rights but do you real want to go to war with you CO over that?

Commander can add to, not take away

If it's not too long u just have to know what to do or what to say. But the stupidest thing in the military the high-ranking can be abusive.

AR 670-1 3-1 .a

“ It is the responsibility of commanders to ensure that military personnel under their command present a neat and soldierly appearance.”

I’d err on the side of the commander on this one due to the general authority granted to them by the army to maintain order and discipline.

#commander’sdiscretion

No, but he or she can make your life miserable until you do. Take a look around your unit. If facial hair is the exception across all ranks, that’s a hint as to the unit norms.