Hi!
Thank you for all the details, I can confirm this is an issue related to the post type filters, and I was able to replicate it on our test server as well. I have managed to construct a fix, but it requires replacing one of the plugin files. Please follow the instructions below:
1. Before starting, make sure to have a site back-up, just in case.
2. Open up the wp-content/plugins/ajax-search-pro/includes/views/asp.shortcode.custom_post_types.php file on your server via FTP/SFTP
3. Replace it’s contents with this.
4. Save the file, and reload the site front-end
Now, the search should properly pre-select the filters passed on to the results page. Let me know if you need help with the file replacement.