[Bishop Edward]

Each regiment would probably have it's own 'name' like the Sisters of Cyric, but the faction as a whole should be named something neutral, or not even have a name as a whole...

My suggestion would be:

Instead of saying something like "I want to join The Dread Unicorn Riders Company", instead have the joining PC's say "I want to train to become a black unicorn rider" and then have the PC division's name changeable by its PC leaders when they get to that point (top rank), or with a placeholder (chosen here?) while NPCs are the leaders, to set the tone.

Would be curious to know where they're located exactly, to suggest a starting name...

As a P.S. I think it would make sense to allow female legionnaires of a certain rank (lieutenant maybe?) to transfer into them, rather than being able to join them right away... since they're semi-prestigious (much less so than Thayan Knights, but still...)