Sorry I missed that…
The problem is the header mini menu and the header image are in the same container, so they are either both over the menu or both under.
So we cannot use the header mini menu for what you want.
Assuming you have the Plus plugin, use an extra menu shortcode to add a Header mini menu inside the Pre wrapper HTML Insertion area and add the CSS below to the area BG CSS+ box
{position:fixed;z-index:10010;top:30px;right:20px;}