How do I keep the page I’m on within the URL when running a search on a taxonomy?

Just change the action attribute of your search form in your theme, for example;

<form action="<?php echo home_url( $GLOBALS['wp']->request ); ?>" method="get">

This will dynamically output the URL of the current request, so that when a user searches, the query string is simply appended to it, rather than pointing back at the root.