I also notice on the theme that they have embeded the SearchWP Live Ajax Search plugin into the theme, maybe this is also causing possible problems?
Could be but unlikely. It is rather that the “load more” request is not an actual search query to the results page, but a custom made query to a custom handler somewhere on the server.
site search be able to use the live ajax search?
Very likely. First I would suggest trying the automatic replacement feature. If the search form is printed via the get_search_form() call, then it should be replaced automatically.
If no luck, then checking the theme customizer/editor might be the way. Most themes offer replacing the widget in the pop-up with anything, and there you could use the Ajax Search Pro shortcode.