Here's how I think it should (and probably won't) work:
As a local faction member, you would have
ranking,
rules,
rewards and
responsibilities, the last 3 would vary from faction to faction.
- Ranking: This would be a new thing - much like the major factions, you could raise your standing within a faction, which would increase your benefits and also increase the level of influence your actions would carry, both within your circle and and in the eyes of others you deal with. High-ranking members would therefore be high-value targets for rivals to turn or destroy, and you might be offered protection when travelling outside your borders. (Note: it's not the same thing as reputation, but it probably could be if that made it simpler)
- Rules: You may be restricted from certain types of activity (trading in particular commodities, piracy against allied traders). You might have to respect some outlandish local customs to maintain your status. (e.g. coming back to home base once a month, not initiating any combat during a special holiday).
- Rewards: These could be simple discounts on commodity trades, new ships, or equipment, or they could be things like "don't get scanned by local cops", "get your local record wiped for a reduced fee at the station".
- Responsibilities: You might have to pay a tithe and/or do odd jobs to maintain your standing in the community. You must represent your faction appropriately outside it's borders.
Now each faction would maintain relations between itself and a fixed number of other factions (fixed to spare the database servers from further punishment). As a member of a faction, you can influence these relationships in accordance to your faction rank as well as your other ratings. Take on enough jobs for a neutral faction, and you might cause your home faction's standing to rise in the eyes of your temporary employers that the two groups become allies. This would cause you to be treated as a honoured guest by your ally, affording you many of the advantages of their membership without most of the responsibilities, as long as you stay within their rules in their borders. This would be tricky when your rulebook contradicts their own, but that's where your judgement would come in - trying to strike up a friendship with those who share none of your values is probably doomed to fail...
Eventually, with enough deals struck and hands shook, your faction's influence could spread across vast areas of the galaxy, and political manoeuvring between rivals would get pretty complex and far-reaching - maintaining that network would not be easy.