Menus are generally very sensitive. The default menu structure is a “ul” unordered list with “li” list elements and also an “a” anchor tag for each menu item. These are inline elements and by HTML validity rules they can’t contain block type elements such as “div”. In modern browsers this is usually no issue, they render these incompatibilities just fine, but it is still problematic.
If you plan to place the search into the menu, I only recommend using the simple shortcode:
[wd_asp id=1]
If it gets rendered it may work with the results hovering below it – or displayed within the page if the results shortcode is present.