Unfortunately the search results page layout cannot be replaced by a plugin, it's not supported on WordPress systems. I'm working a possible get-around solution for a future release, but it's very problematic, and I have to constantly consult with core developers with this case. The plugin can override the results to display on the results page, but not the page layout.
The results page is fully controlled by the currently active theme, usually the search.php file is the handler in the theme directory. The fastest way to override this layout, is to create a child theme, and create a new search.php file – but that requires a significant amount of coding.
A possible get around would be to use only the live search feature on the page you have linked, not using a redirection to the default results page: https://i.imgur.com/Lcz1cY2.pngBest,
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