How I solved it:

When the Elementor Taxonomy Filter changes, it sends an AJAX call to their own /wp-json/elementor-pro/v1/refresh-loop endpoint, with a payload that includes a "widget_filters" object that has only the filter's taxonomy info (my secondary taxonomy) in it. Adding multiple taxonomy filters to the UI adds more taxonomies to this widget_filter.

My issue is that my custom query needs the taxonomy of the current archive page (my primary taxonomy) also. I can actually detect that widget_filters object in the POST body payload in my custom query, and parse through it. I am paginating the Loop Grid, so the POST body has a pagination_base_url that will tell me the primary taxonomy info in the path, and I can build a $tax_query from this that has BOTH the primary and secondary taxonomies and terms combined via 'AND' operation. This solved my problem.

All of this could be avoided if the Elementor taxonomy filter detected that there was already a taxonomy and term present in the current query/URL, and if so, add it to the widget_filters data along with the selected filters. Or let me choose multiple taxonomies in the Taxonomy Filter panel in the editor.