I think the 'GalCop' revivalist movement (quotation marks added because no major faction is likely to recognize it as the continuation of GalCop) would make sense, particularly if it can be pinned down to a specific region with abnormally active pirates/a lot of Imperial-Federal proxy wars in its recent past. I mean, there are still plenty of people out there who claim to be successors to an organization they have no real ties to. Why the heck wouldn't a group of independents resurrect the GalCop name? It's not fundamentally different from making up some random name, except it's transparently trying to hearken back to 'the good ol' days' and a somewhat laughable threat to others ('oooh, GalCop found a time machine, better pack up our proxy wars then').
GalCop in this sense would be one of the confederations of independent factions that crop up from time to time in response to the core worlds domination and lack of respect for their sovereignty - typically these don't last very long, but this is the kind of movement that led to the formation of the Alliance so anything's possible. The only thing I really have to say is location is key! If you build up the lore of the region GalCop aspires to occupy in a way that's publicly accessible, people will be much more sympathetic and invested.
The single biggest problem I suspect you'll face is justifying why the various governments supporting GalCop aren't angling to join the Alliance - I mean, they're nearby, ideologically friendly (they're basically the closest thing GalCop has to a successor, though admittedly not that close) and they have the resources to make what GalCop is trying to accomplish a reality much faster and more reliably than a bunch of independents who don't quite trust one another. We had a similar deal with the Coalition, which we're attempting to resolve by emphasizing how it's just diplomatic maneuvering and not a proper attempt at a state, just one demagogue persuading various independent factions to cooperate, and by stressing that we're accepting of government types that the Alliance most certainly would not be. Perhaps some reason for GalCop space to resent the Alliance could be found?
Best of luck with the games!