I haven't done it, but it might work to use a plugin connected to OnManagerPageInit that checked the action and injected the JS if the action is the search page.
In my install, that would be:
I think more correctly it would something like one of these (but I'm guessing):
if ($_GET['a'] == $modx->action['site/search'])
if ($scriptProperties['action'] == $modx->action['site/search'])
The search Tpl is just this:
<div id="modx-panel-search-div"></div>
so modifying that isn't an option.
The JS is injected in this file:
manager\controllers\default\search.class.php