I think I would leave the tabbed layout as it is currently, as it gives a nice separation of the results.
Then, instead of the search on that page, I would simply replace it with ajax search pro. It might require to change it within the theme, depending on how it's designed. If the automatical replacement does no effect, then there is the PHP search shortcode for that, and in most cases the file the search form is placed is the search.php in the active theme directory. Unfortunately there is no exact tutorial for that, as it varies greatly for each theme out there.
Then, assuming ajax search pro is in place, I would change the behavior to automatically act as a simple search when the user hits enter or the search icon, but keep the ajax functionality for automatic quick suggestions if the user just types in a keyword: https://i.imgur.com/Fa7Qoby.png
I believe this is the closest way of merging the two together without getting into coding and possible conflicts.Best,
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