What's a league name? Wikipedia is one of the sources I used to decide between Imperial and imperial, although I've since seen some inconsistencies in their articles. If I use Imperial, that suggests to me that Imperator (like Emperor in Star Wars) should also be caps. If I use Imperator, then I probably should use Imperatrix and Imperial Heir. That leads to Royal Regent and Crown Prince, which leads to Princess. The regent rules an entire planet, but so do governors, but I've never seen governor capitalized. Etc. Let me know if I'm misunderstanding you. I prefer using caps for many of these, but it breaks down as you follow the chain of logic, above. President in caps is unique to the US. At least one of the style guides I referenced says not to capitalize it.
I adopted my own style, which is to lowercase informal use of titles and caps formal ones. So you get "Apollo is the imperial heir," but "Apollo is Heres Imperialis of the Imperium Romanum."