This is a bit of a tangent, but I like to think that the species that have been absorbed into the Empire ARE subordinate to the Klingons. Player characters are the notable exception. The KDF tutorial should have a story or mission that explains why the non-Klingon characters are exceptional, and given the honor of being treated as equals while others of their species are not.
While I conceed that this would make some people uncomfortable (likely bringing up the specter of racism and discrimination), I believe that if this aspect is handled properly, it could get people to start thinking and talking about the evils of prejudice and perhapse lead to better understanding.
I had other ideas for justifying the addition of non-KDF uniforms, but I think I should wait before I suggest a reason(because that "justification" could be considered controversial). All I will say is that Klingons are not human, and what we find horrifying, they might think is normal and acceptable.