Hi Matthias,
The issue is not related to the search, but that the page builder section where the plugin is included, hovers over the menu.
There might be a manual option to set the z-indexes somewhere, but you can also try this custom CSS:
[html].elementor-element-801e641 {
z-index: -1 !important;
}[/html]
Please note that this might not be the best solution, as it is not directly related to the this plugin.