If you think the Klingons, even the smooth-forehead variety seen in TOS, would allow someone else to do the fighting for them against their enemies, then you are not talking about the same Klingons that have been shown on screen over the decades. They have their covert sides, surely, but they never send in some disohonorable mercenary alien to do the job of a stealthy warrior. That has been their downfall more times than I can count, most notably the incident with the Tribbles that had such a long lasting impact on the Empire.

Now, Romulans I would expect to delegate out their dirty work. Cardassians might as well, if they didn't enjoy most of it so much. Neither of them follow a warrior's code of honor because they value other things much more than honor. Romulans value privacy and victory above all else. They are prideful and vengeful. Cardassians are pragmatic and self-centered, but value family above even themselves, usually. They are coldly cruel and efficient and will do whatever it takes to achieve their goals.

Klingons are passionate. They are violent. They are warriors through and through. They are also honorable, and if they see that you are honorable, they will respect you. It was decades of conflict with the Federation that taught them about the strength and honor they could find there, especially in the Humans. That's not something that can be learned through proxy wars or cold wars. It's something you only see in battle against the enemy. How they fight, how they act in defeat, how they act in victory, and whether there is any honor in it.

Kirk and crew saving the Klingon delegation from assassination in ST:VI showed them the honor of a strong race who had won a victory through attrition. The Enterprise-C being destroyed in battle with the Romulans to defend a Klingon civilian colony during a time when there was no alliance between them showed them the honor of the same race when there was nothing to gain and possibly no one to see it or sing their song. It was mutual respect from years of war, and mutual admiration from the intermediate years of peace that grew the alliance between the Federation and the Klingon Empire. They still have major differences, which is why the Empire never joined the Federation, but they are closely allied enough that it takes extreme circumstances to break down the diplomatic relationship between the two governments. Even after the events of the early Dominion War, the Federation and Klingons quickly patched things up to resume their peaceful alliance to battle the Dominion.

A random captain firing on a Federation vessel who he believed had tried to attack it through subterfuge would be acting in accordance with his protocols as a warrior. The Federation values peace enough that even losing that one ship to such an act would not cause them to resume open hostilities, or even abandon the alliance. The Klingons value honor enough that they would take the accusations of the Federation seriously and likely punish such a captain quite harshly if he did such a thing. Foolish captains exist in any fleet, as do ambitious captains. In Starfleet, they can be overruled by regulations, even on their own ship. In the Empire the crew, especially the second in command, can challenge the captain's decisions by questioning the honor of the action or challenging the captain for command through personal combat. Unless you have a bridge crew that is completely without honor or courage, it would be exceedingly rare for such dishonorable actions to occur.