The extension does not successfully change the default search engines to Kagi in the Vivaldi browser, and it is not possible to do manually (in the entry that the extension added) either.
I have determined the likely cause to be the @ character in the "nickname" field, which the extension seems to try adding by default. Vivaldi does not allow @ in that field, so my guess is that that is what is causing the extension to not work properly.
The workaround is of course to add a completely manual entry, which doesn't really help with usage in private browsing.