Yog Sothoth said
Your faction sounds like it doesn't have too much freedom and individual choice. Also this a galaxy we're talking about so it is highly possible for there to be a galactic crime syndicate, and I think it would offer a lot of interesting aspects to the rp.
It being a galaxy (which is questionable as the GM has said that at maximum all our combined civs will cover about 10,000 stars) has nothing to do with the liklihood of there being an organized inter-faction underground.
For something like that to be plausible you need two things: high inter-connectivity and more or less unregulated civilian access to spaceflight.
The first point fails, because even basing it solely off of the current completed sheets, factions are trending towards being fairly insular; they exist as largely self-contained bubbles of one or a few close-knit species united under a larger political body, with areas of empty space between each other. Sure, within Redspace or Graal or individual Yul worlds you could probably find a criminal underground network, but it would be exclusive to those locations and have no functional reach beyond them.
I mean, this is the same reason that spies aren't very plausible in this sort of setting; individuals or organizations from one faction are going to be very obviously foreign if they interact with other societies. A Graal could never infiltrate Redspace because Graal are rare and stand out and would be subject to automatic scrutiny.
The second point is also likely to fail due to a lack of the first element.
In a setting like, say, Star Wars, where species can generally be found anywhere and there aren't many factions, sure, underground crime ring works. In something like this? It's rather unlikely.