Add a checkbox to voluntarily disable the error message completely. If the browser is not running and the warning was disabled in the configuration, the search page should stay completely untouched by the plugin. No "Joplin search notes" block at all.
It sounds like just adding an option should be simple, but maybe later, showing a non-blocking error doesn't feel like it will have much impact.
Let us modify or even style the "Joplin search notes" title and the whole block.
I'm not sure about that, maybe some kind of custom css style is supported.