I took a quick look at the code, but on the event handler for handling the selection of checkboxes can you at least get something to trigger so you know that part is working. For example log out what the new request is along with the modified query results.
Also can you mock it up to give you a custom query from the start by hardcoding it like one of the custom taxonomies have been selected.
Just thinking of a few ways to begin troubleshooting.