Currently as screen reader user you're exposed to 2 buttons labelled "Search", but <button class="search-toggle"> is only triggering the search input visibility for GUI users. Therefore it should feature aria-hidden="true" attribute.
<button class="search-toggle" aria-hidden="true"> … </button>
In the original description probably due to a lack of sleep ;) I misstated that aria-disabled is the attribute needed, but with the input and a corresponding button side-by-side, actually aria-hidden is resolving the issue completely here. That has also been implemented in the corresponding pull request.