Well, the best solution would be not to use any plugin to create taxonomies at all, and if possible, then not to use custom post types as taxonomies either. Taxonomies can be registered very simply in the functions.php file in the child theme directory: http://www.wpbeginner.com/wp-tutorials/create-custom-taxonomies-wordpress/
They explain the manual method as well, which I highly recommend. It removes a huge overhead, as you don’t need a plugin for it.
If you plan to keep it this way as it is now, let me know, I may be able to construct some sort of small custom code that would remove the original search filter code and re-filter the items after search.Best,
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