Klythe
Probably you can look to Corax's behavior in the bar on Station K-7 for how Klingons would behave towards the Federation. They'd likely be looking to start a fight, but they would want to provoke the Federation to attacking first, just in case the Organian's intervene they could claim to be the injured party.
I like that, It definitely fits into the TOS klingon behavior pattern, but still gives my players a reason not to start disintegrating each other at their first encounter.
I can have the Orgainan treaty (and possible intervention) hovering in the background as a largely unknown dynamic, this allows the Klingons to be insulting and antagonistic without a total war breaking out.
Thanks