I think the crux of the matter is whether this GMG is really a faction in "hardship" or a new faction altogether.
If the latter...then technically they should go through the "becoming official" process all over again (which they probably would make no problem). More bureaucratic work for the admins, but really doesn't affect current GMG at all aside from losing official faction perks for a short period (which they didn't much seem to be using anyways).