In "A Matter of Honor", Klag similarly suffered from a dishonorable kinsman, but unlike Kurn, he certainly didn't wallow in despair. Then again, Klag only had a dishonorable father - Kurn and Worf suffered from a collective debasement of the entire House of Mogh. Apparently, being captured alive was less of a crime than selling out an entire planet to Romulans, so the punishment was narrower in scope, too.
In any case, Worf generally benefited as a character whenever he was a badass or an alien, i.e. when he murdered or insulted or cheated or stonewalled where a human would have showed compassion. Trying out all the possible options and then, just before end credits, nevertheless sending Kurn to Sto-vo-kor would probably have worked out very well indeed. Especially if all the clever and humane machinations of the rest of the characters were setting up a happy ending for Kurn, but they also
made it possible for Worf to give Kurn an especially honorable end.
Say, perhaps Kurn was going to be memory-wiped and sent to a safe fate A - but Worf took advantage of this and derailed Kurn to a glorious fate B, perhaps a suicide mission under "false" identity or something.