Asajj - Sith
Fulcrum - Phoenix
HK and Visas - Old Republic
Veterans Han and Chewbacca - Resistance
Surely none of these characters would make the factions overpowered. The only exception would be maybe Fulcrum (just make her Unique non-shareable, like how Chopper's is), or maybe asajj.
Edit: If Old Ben can get a Rebel tag (the only thing he did was make Luke a Rebel...then he died), then surely at least one of these tags can happen lol.
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Fulcrum does fit in with Rebels but did she go on many missions with Phoenix squadron? A few to be sure so she is perhaps deserving of the tag.
Visas was never OR though she is from that era. HK is a bit tricky, really depends on whether JKR made him (making him OR) or Darth Revan (who had rejected the OR and embraced the Sith).
Veteran Han was never Resistance in the movie though the books imply he was a peripheral part at one time. Veteran Chewie could become Resistance by episode 9 but now serves merely as Rey's chauffeur.