Hi,
The built in compact layout mode supports a sort of ‘opening-closing’ mechanism, which is basically done by changing the search box size.
To have it opened below, you will need a custom script and HTML solution. I have seen some themes have something similar, but it always depends on the exact output structure. Most of these are built as-is, so you can’t replace them with another search bar.
I think the easiest solution would be to add a separate magnifier element (as an image icon or SVG) to the desired location, as well as a hidden container, which would include the plugin shortcode executed. Then a custom script would detect clicks on the custom magnifier element, and position and show/hide the container, that includes the plugin shortcode. For custom jobs, I usually recommend this service – they are dedicated WordPress developers, this should be a fairly simple task for them.