To be frank, I think your method of taking over the entire lore of Phoenix is very disrespectful, moreover without any consultation or notice with the previous members.
Think of it as stealing a image on the internet, edit it to your liking and then claiming as your new work. Of course the original members who did all the effort into building the faction as it was, would be angry with your actions.
In addition, the majority of the Phoenix group left SIrius, along with all their home ships and support fleet. Yes, there are remnants members across Sirius but they are feeble group. Having suddenly re-gained all the influence across O-74 instantly seems out of proportions, not to mention you are retconning the existing Phoenix lore eg-by revising the situation / stating more powerful than you necessary are.
Having said all of that above, I wouldn't mind seeing a splinter group deriving from the Phoenix lore, but you should really drop your status as the Phoenix in the process. Perhaps be create new / different (or better) group, by starting off new as a new page in the book instead of revising the boring past?
PS - Just before the Phoenix left SIrius, they had issue with order / bounty hunters / zoners relations. Maybe you should focus building this issue / balance into your new lore as a different faction ?