I think the issue is character slots. as most people don't have excess they can only create one Foundry toon. That is usualy a FED one.
Thats not really an issue because your missions arent tied to your character. That means you can delete your current character and make a klingon just to make a KDF mission, then delete him when your done and make a new fed again. Any missions you have made remain no matter what you do with your test characters.
I have two character slots on the foundry. Separate from my actual characters. One for Fed and one for Klingon...Don't try to create both Feds or both Klingons. I tried and had to delete both because the 2nd Klingon or Fed inherited the opposite of the first character. Was weird logging on to my Klingon character with a Fed interface... lol