No. Just no. "Cultural clothing" is an important part of a species' unique style and, in my opinion, does a lot for the atmosphere in that it actually makes it look like in the shows.
You know very well what this would result in, and personally I can do without every 2nd female Klingon and every single Trill, and every single "Alien" Human running around in Orion bikinis. It's bad enough that we have Alien Humans at all - I still see this as exploiting the system.
Please don't turn Qo'noS into a second ESD. Keep the zoo with the Feds.
Short versiont: yup.
There are a few things to consider:
The KDF as it is now is essentially an allied force, like the Warsaw Pact or NATO.
Meaning they are not one unified force in the sense the Federation is.
The analogy is also fitting in the sense that the largest country has overall command of the alliance in question, just like the Russians had in case of WP or the U.S. has in case of NATO (Supreme Allied Commander is always a U.S. officer).
So while I think it is okay to have a certain dominance of Klingon ships because it's the kind of ships probably produced to a greater degree (simply because the Klingons have more shipyards than others) and also possibly sold to other nations to fill up gaps in their forces (just like the U.S. and the Soviets sold military equipment to their allies to supplement what they had) each part of the faction as a whole would have its own clothing.
Just like the individual nations of WP and NATO had/have their own uniforms.
While we inevitably run into problems with the "aliens" because they'd probably end up as some kind of "Klingon foreign legion" wearing the uniforms usually associated with the Klingons and not something specifically "alien" (except when you use some tricks in the editor) the others like the Gorn, Nausicaan, Orions, Letheans (in othet words the "named non-Klingons") should have clothing of their own.