brn I find a way to reproduce this.
I am using Hardened Firefox with Narsil/user.js with some custom settings. If you need my custom settings, maybe I can send you. I haven't tested on a normal Firefox Browser.
In non-private window, the extension works just fine.
In private window, the Kagi extension shows the token is being set:
but, Kagi is actually not logged in.
I have tried to set the tyoken manually in the extension, I don't know which one below is the token, so I tried both:
But, when I click "save", it just keep stuck in "Saving...":