I ended up doing the whole Galatia academy questline after resolving the PL colony crisis by joining the league with 2 "strong arguments" for Reynard. Doing the parts of the quest involving the League felt sort of jarring and weird, because PL officials were treating my character like some no-name spacer, despite my Commission, max rep with the PL, and being the leader of several colonies within the league! On top of that all, I had a strong relationship with the literal prime demarchon. My point is that in these scripted interactions throughout the game, the players reputation and the in-universe clout there name should hold isn't represented very well, and not just for the league.
ps; as a side-note, it would be cool to have some sort of internal political system within the league which the player could participate in after joining.