I see this issue on Windows 11 and macOS 26.5.1.
Launch a Brave instance. Open a regular window.
Install both Kagi Search and Kagi Privacy Pass extensions.
Set "Authenticate via Privacy Pass" to "Off".
Highlight a word. Right click and select "Search Kagi for <word>".
Observe that the search resolves. So far, so good.
Set "Authenticate via Privacy Pass" to "On" or "Incognito only" - these have the same effect even if you aren't using an incognito window.
Highlight a word. Right click and select "Search Kagi for <word>".
Redirect fails with ERR_BLOCKED_BY_CLIENT.
Redirect ought to succeed.