Update! I've gotten it to sort of work by making these changes to the code:
<script>
$(document).ready(function() {
$(".tweak-header-search-style-icon-only .Header-search").click(function() {
window.open("https://actionlab.strongtowns.org/hc/en-us", "_blank");
});
});
</script>
The spyglass does open the "Action Lab" website now, but in a new tab... Meanwhile, on the old tab, the user gets directed to a blank page (our old search page, now deactivated). Any ideas on how I can get the link to open in the same tab, instead of opening a new one?