Ok - if you want to drop the AJAX then you need to add the Search Button to your "Search and Pagination" Section to that should help to submit/trigger the sorting. Once you add the search button let me know and I will see if there is any requirement to adjust the code after adding search button.
We already have 2 Search and Reset buttons in the Search and Pagination section. One set to appear at the top of the filter critieria and another at the bottom. We know that the user has to manually click the Search button to refresh the results currently.