Search suggestions with private search engine

I setup a personal SearxNG server behind authentication, which needs cookies to validate the session. When using normally through the browser, it works, but when trying to test the search suggestions on the address bar, I was getting access denied on the server.

I tracked the issue due to Firefox not sending cookies to the search suggestion endpoint, and discovered that it’s intentional:

Added in

I understand the fear of sending cookies to third parties, but that effectively makes my search engine less effective.

Say, what alternatives do we have to add search suggestions to Firefox with a private search engine?