I have had an odd issue for a while: Kagi Privacy Pass would work perfectly in regular browser windows, however as soon as I launched a private window and tried to do searches, I would receive the Chrome error ERR_BLOCKED_BY_CLIENT for the extension’s chrome-extension://mendokngpagmkejfpmeellpppjgbpdaj/pages/redirector URLs in Vivaldi.
The source of my problem: I added Kagi as a custom search engine with my token in the URL. This token was also being passed as part of the redirector, causing the request to be blocked.
The fix was just as simple: Add another instance of Kagi as a search engine, however without the token parameter. Enable this second instance as a search engine for private windows.
Problem solved.