I've attached government owned NFA to personally owned NFA/GCA firearms (and vice versa), but I'm not military. I think it's up to his chain of command as far as what is authorized use. As far as federal NFA/GCA is concerned, if he's authorized by the owning government entity to be in possession of NFA items for duty related use then he may possess them anywhere (above CoC issues notwithstanding).
It's correct that military weapons are not registered with BATFE, and never have to be.
I know SEALs and other high-speed types draw (sign out) their issued weapons all the time, and take them to civilian training classes, on authority of their CoC.
It is my opinion that the pistol grip (even though it's part of the launcher) would render the host pistol an AOW, but my opinion doesn't matter. I'd ask BATFE if the M320 may be mounted to a pistol. That's all I'd ask them. There are civilian-owned M320's, so that's a logical question.
Really, I'd just find a rifle to mount it on (assuming he's allowed to have the thing in the first place).
If he's authorized to be "training" with it off-post, and he slaps a host SBR together temporarily to use in said training, it probably wouldn't be a NFA violation because the firearm would technically be possessed under the authority of the DoD (and therefore not required to be registered) but I haven't seen any case law on that situation. If that's legal, then a temporary AOW during training would be legal as well.
I guess the "dumb question" to ask here would be... If your buddy can draw the M320 for use off-post, why can't he draw an M4A1 as well?