:::spoiler an interpretation

It might make more sense if you replace Joshu's actions with "doing a jerk-off motion." He's treating the discussion as unserious, as well as treating himself as unserious.

Nansen shows up looking to start an argument in bad faith, no doubt with all sorts of rhetorical talking points prepared designed to catch the other side in a gotcha. Attempting to engage just takes you into his prepared talking points and responses, while evading the question or refusing to answer makes him feel like he's "won," and leaves his words unchallenged.

The only thing you can really do is to take him off script and take the wind out of his sails. Nansen was prepared for the possibility that they'd answer, and prepared for the possibility that they'd refuse to engage, but he didn't have a plan for what to do if someone put a shoe on their head.

Also, by treating himself unseriously and humbling himself, Joshu removes Nansen's motivation to prove his superiority and the superiority of his house. Joshu essentially "shows his belly," and says through his actions, "Bro, I'm just a silly little guy, if you wanna kill that cat, that's all you." Nansen's internal justification for killing the cat could only be that he wanted to knock the other side down off their high horse, and so by abandoning any sense of dignity he would remove that motivation, and the cat's life would be saved.

By both having the wisdom to understand Nansen's thought process, and having the humility to put aside his dignity for the sake of saving the life of an innocent being, Joshu ironically displays a better grasp of Buddhist teachings than Nansen, regardless of whether Nansen could beat him in a debate through wordplay and rhetoric.

Those koans were stories written by Mumon. They weren't all based entirely on real events. Maybe Nansen gave a dharma talk where he said he would kill the cat, but to actually kill it would go against the precepts. Highly doubtful that actually happened.

The point of the koan isn't surface level though, and if you expect to be able to extract a lesson from a koan by one simple reading, then you're approaching koans wrong to begin with.

Mumon's commentaries might help guide your understanding, and perhaps secondary scholarship like a translator's notes or whatever. But unless you're viewing it from inside the tradition, it's likely to seem obscure/perplexing.

But unless you already understand what he means by "say a word of Zen," and why Joshu placing a sandal on his head was taken as an acceptable answer, then this story will elude you no matter how you look at it.

Ah yes, enlightenment