I fixed this by adding the following code in the header-4 area just before the cart:
$header_output .= '<li class="menu-search parent"><a href="#"><i class="icon-search"></i></a>'. "\n";
$header_output .= '<ul class="sub-menu">'. "\n";
$header_output .= '<li><form method="get" class="search-form" action="'.home_url().'/"><input type="text" placeholder="'.__("Search", "swiftframework").'" name="s" /><input type="hidden" value="product" name="post_type"></form></li>'. "\n";
$header_output .= '</ul>'. "\n";
$header_output .= '</li>'. "\n";
I think this should be added in the next theme update automatically – no reason why header 4 shouldn’t have search field, in fact in mobile responsive mode the search bar does show up out of nowhere.