Cryptic is likely to be introducing FvF into the game soon (it's currently on the test server). Let's suppose, for a second, that as a consequence the KvF queues dry up, making it even more dificult to level/advance your Klingon character. What would you do then?
1) Nothing. Rely on KvK and Nebula missions.
2) "Re-roll" Federation (or simply stop playing your Klingon toon in favour of your Federation one).
3) Quit the game for good.
4) "Take a break" from the game until some favourable solution is implemented; i.e. Klingon PvE, more/better Klingon PvP.
5) None of the above (please do explain).
Cryptic is dropping the ball big time if they continue making FvF, FvK, and KvK queues separate. The proper mechanic is to queue for a map, and have the system give you 5 opponents of whatever faction is available. If there are 5 Klingons in the queue, you fight them. If there are 5 Feds, you fight them. It would make no difference whether Feds or Klingons are queued, as long as there are 4 others from your faction (to group you with) and 5 more on either faction (to fight against).
To answer your original question, supposing Cryptic DOES fail horribly and leave the queues separate, AND it somehow leads to a PvP drought for Klingons (I disagree that this would happen, but that's not the question) I'd just play my Fed as a PvP character.
I would far prefer it if Cryptic made the queues such that if 10 or more Feds are queued and no new match has started in 10 minutes, they are then put into FvF match. If not, it's FvK first.
So basically, it works like this.
There is 1 queue for a certain map(say Solar Wind).
Klingons and Feds queue as per normal.
If after 10 minutes, that queue has not spawned a match, check the Fed and Klingon population queueing. If there are at least 10 Feds and less than 5 Klingons, start a 5v5 FvF. If there are at least 10 Klingons and less than 5 Feds, start a 5v5 KvK.
Within 10 minutes, if there are 5 Feds and 5 Klingons, just start a normal FvK.
Klingons live and prosper through PvP now. That's how their gameplay was advertised. You cannot just yank the rug out from beneath their feet now.
10 minutes is an unreasonably long time to wait, and it accomplishes very little other than arbitrarily punishing Federation players.
If faction-agnostic queues fire off immediately when 2 usable groups of 5 are present, then they should. Assuming that there are a much greater number of Fed players than Klingon players in queue (generally the case), the Klingons will almost always end up with quick FvK queues anyway. If anything, the complaint should be that KvK will never happen... but who really cares about KvK in the first place?