Great timing, same issue, also on Firefox. Also seemingly mitigated by re-saving the token (with the same complains as @RoxyRoxyRoxy).
However, I've noticed this is a recurring issue - but I haven't tracked down the interval (might be related to Privacy Pass, started happening around that time) - but the extension will randomly fail to work on a new private window (token is still valid).
I don't think it's related to FF restarts - maybe it's related to Kagi's token being removed from your cookie storage when the private window sandbox closes (if you logged in like normal to the private window)?
