li.header-dropdown-toggle won’t work, because it’s not unique to the Quick Messages icon. It’s shared with the search and hamburger menu as well.
The first element that’s unique to Quick Messages is inside that
<a> element which has an id
#toggle-messages-menu. As you already saw, it’s the
<li> elements you want to rearrange (as do I).
<li> element, or add a setting to the plugin.