The extension tries to remove the token from the URL before the request is sent to the server because otherwise the search wouldn't be anonymous. It succeeds when you search from the address bar, but it gets stuck when you search from the context menu for reasons presumably involving Firefox's security access checks not liking that we're redirecting from whatever originates the context menu search request to a trampoline page that lives inside the extension, or something like that. I think it's a bug on their end so I filed https://bugzilla.mozilla.org/show_bug.cgi?id=2033905.
The immediate solution is to set a search URL that doesn't have a token parameter, like the one you get from the Kagi extension (which takes care of auth another way). Hopefully it'll be fixed more generally soon, either on Mozilla's end or possibly incidentally by upcoming changes in the extension.
