I'm sorry, but the Klingon Empire is most assuredly NOT part of the Federation; if they were, how could the two go to war with each other in DS9? They are allied, but this does not imply that the Klingons are members of the Federation. If they were, they wouldn't have such a huge star fleet, for one thing, nor could they launch wars of aggression against Cardassia without Federation acquiescence.
Secondly, the exchange is hardly unusual in Federation-Klingon diplomacy. It was said by George Kennan, former secretary to the US Ambassador to Moscow - and later the US National Security Advisor, I believe - that the Soviet Union only understood negotiation from a position of strength. If you asked them nicely for something, they would promise to give it some thought, then do nothing. But if you simply told them what you wanted, and backed it with the threat of force, they would probably give it to you. This happened when the US threatened to intervene in Iran in 1946, the US refusal to allow a joint occupation of Japan, and the Berlin Airlift. The US did not apply pressure over Poland, so the Soviets therefore didn't budge.
The same is true of the Klingons. They are clearly a militaristic culture, and negotiating from a position of weakness would only invite their contempt. Threatening them, even non-seriously, was the best way to ensure their co-operation. In "Unification I," Captain Picard asked the Klingons for a Bird-of-Prey to infiltrate Romulan space, telling Gowron's representative "you will have our gratitude. If you don't supply the vessel, I'm sure others in the Empire will, and they will have our gratitude." This is a very blunt threat to back another side in the fluid politics of Klingon society, where Gowron had only just emerged Chancellor after a bloody civil war.
Such threats work both ways. Gowron routinely refused to consider returning Cardassian or Federation territory even after the Klingons allied with the Federation against the Dominion. Martok expressed contempt for Cardassians and humans even though he stood amongst the corpses of Cardassians whose rebellion against the Dominion had ensured the Federation Alliance's victory. Worf's brother, Kurn, threatened to kill Riker during "Sins of the Father." This is simply how Klingon society seems to treat negotiations.