Hi!
Thank you for the details, it helps us a lot.
Okay, so I have tried to open the linked page, but unfortunately I am not seeing anything there.
From your description I’m guessing that the data is stored within a tablepress table (?), and then the table shortcode is used on that given page. In order to search and return the contents of that page, you will have to use the index table engine.
One thing to note, is that the plugin cannot return parts/lines of a file as separate results, that is way beyond it’s capabilities. The results have to be separate WordPress objects, like Posts, pages or custom post types, or taxonomies etc.. Filtering table/file lines requires a custom made solution, as each implementation is vastly different from one another.