There is an issue with clothing options. The races from each side isn't programmed to wear outfits from the other side, with the exception of Klingons and Joined Trill. Probably wouldn't be hard to program all races to be able to wear everything, but certainly not high priority.
Actually, they absolutely can. The issue is that the costumes don't mix and match.
Feds use body sock costumes whereas Klingon costumes add bulk. (Although given how few Klingon costumes there are, I don't see why they couldn't be reworked to be body socks with attachments, ie. making vests an armor layer, etc.)
As long as Fed and KDF costume parts don't have to mix, it works fine. Which is why both sides can wear Omega Force armor.
Even then, the only real issues are with collars and boots.
Here's the worst conflicts I can find that you'd get: