Klingon ships aren't by default 'superior'. Klingons are better in PvP because they have more experience. They'll also be better on Fed side, with Fed ships.
Allowing Feds to fly Klingon ships would still be a show stopper, because no-one would play KDF anymore if *everything* was on Fed side. ("Everything" = content, selection, extras, species, ships, abilities).
I think maybe they should have non-aline fraction or they could put you on diff ship for that mission. At end of the mission, I'm not sure if you should get to kept it. You could learn about there weapon, also give chance to play as that race without build creater.
Or could just sell you one. You then get go thought a unique mission, they setup for you. That I like the most I think,