FYI this issue is still ongoing. Would be great if it could be fixed.
Interestingly, I could get it to work by disabling the extension and manually setting the kagi seach url with the session token, but then I was losing the convenience of clicking the extension button to summarize etc. So ironically, if there was a toggle in the extension that would allow to disable overriding the search engine, then I guess that would work… But as it is now, enabling the extension automatically takes over search engine selection.