You need to put it on the other menu (the one at the top)
When that is done, you could add the rules below to the theme Global Custom CSS Rule box
.is-desktop .menu-search {margin-top:-30px;}
.is-desktop #nav-primary .wvrx-menu-container {top:98.2px !important;}
.is-desktop #nav-secondary .wvrx-menu-container {top:30px !important;}
body.is-desktop {margin-top:157.4px !important;}
.is-mobile .menu-search {display:none;}
If you increase or decrease the 30px value of the menu search top margin, do the same with the same amount to all the other top and margin top (ie: -20px ; 88.2px ; 20px ; 147.4px)
Last rule will hide the search on mobile
If any issue post back with rules in place