I'm with MB.
If the CO is behind the CMC, don't dig a hole for yourself....if not, do what you want.
He's a CMC from the shoe community, right? Does he have or wear a green flight jacket. If you must comply, become his personal uniform police nazi and look him over all the time for "violations."
I'd get a piece of black velcro put on and stick a nametag there when at work, remove and wear out in town.
Here's mine and I wore it this way for the last 6 years of my 20 on active duty.
There are only three aviators, with me being the only tailhooker. Green flight jackets are banned. We had flight suit friday for 1 day before that was no joy. Its aviator extinction!
It was a good fight to explain that rank is not required on patches for the jacket and that we are actually issued them!