Storm is 100% correct and I like the answer, a real answer.
Jadzia is an exception. Just because one joined trill was great friends with some Klingons doesn't mean they all were. In fact DS9 largely hinted that Jadzia was the exception.
Sure, some Joined Trill besides Jadzia might be great friends with some Klingons but Joined Trill are already rare and I doubt there's more than a handful and none would likely be more respected than Kurzon was.
Joined Trill being on the side of the KDF has a precident, but it has never been depicted. Considering that Joined Trills (and this part is canon) consider the wellbeing of the symboint before the host, it makes little sense to join an empire that sends their troops out to die regularly. Starfleet is an exploratory corps, STO just turned it into a millitary, the KDF was pure military organization.
Dax, well, being Dax and having 2 seperate hosts involved heavily with the Klingons, it was only natural to go that route with the character. But on principle, the whole "protect the symbiont" thing flies in the face of being part of the Klingon forces.
Well i second BigRed's idea....bring extra plastic cups. And not the stupid small clear ones.
But i have to say it is perfectly reasonable that other joined symbionts have previously formed relationships, or made blood oaths with other Klingons. Good reason for them to join the KDF. I don't recall any canon that says this can't be possible, but I could be wrong. A Blood Oath is a manner of honor so they may feel bound by that to join the KDF.
as far as putting the symbionts well being in front of the host....well who says joined trills don't break rules.