It looks like the following code, the z-index works differently on mobile:
#masthead:not(.menu-absolute){
z-index: 2000;
position: relative;
}
I’d suggest you put the HTML for that button in the same line as the logo and mobile menu then hide/show it when you need to.
It would placed under this HTML
<div id="logo-container-mobile" class="col-lg-0 logo-container middle">