I was surprised that my Fed Klink didn't come supplied with a sharp razor
Why would you be surprised by that? Back during WWII there were plenty of American soldiers who were of German ancestry or even German-born fighting for the United States. Pretty sure none of them came pre-equipped with their own Luger pistols.
Obviously it depends on what your character's background story is. Maybe your guy came directly from Klingon controlled space or has been a Federation citizen for several generations. Either way as has been stated there are ways to get a Bat'leth in the game regardless.
I used the ones from DS9 until I hit Admiral, then bought a Mk-X from the exchange for under 10k I think (which is not a big deal at Admiral). It's very effective in PVE combat, even as a science captain with no martial arts skills. It does more damage than most rifles and bypasses shields.
Compared to an equal Mk phaser, Bat'leth do a lot of damage.
How come Federation don't have their own version of a melee weapon?
Where's that switchblade sword Sulu has in the new movie?
EDIT: And how come we can't unsheathe our Bat'leth at random, like how people randomly run through Starbase with their phasers out. I would love to walk around waving a bat'leth like some people wave their assault rifles.