Random aside since we're talking about LMGs, but I think an interesting way to balance it (not really appropriate for the current game since it would be a major change) would be if LMGs got an overall DPS nerf, then had their DPS restored when in cover with a mechanic similar to Tommie/Ostruppen cover buff. Even with the on the move lack of firing I feel like they're still overall too good as a universal upgrade you want to give to every damn squad.
The problem with that is it would hurt Grenadiers more than anyone else, because Ostheer cannot create cover under normal circumstances. It would benefit USF the most because Riflemen can wield LMG's and create their own cover.
Also doesn't make sense visually since out of cover model goes prone to fire, but in cover model stands.