Hi Ross,
I’m afraid no, no change is needed on that code. If you are using WooCommerce, then it will automatically do it’s thing on the taxonomy archive page. Note, that that works for archive pages with the live results (not elementor results).
Might be a bit more simpler to just create two search bars, configure each to return results from that specific category only. Then you can create two elementor pages for the results – one for each search bar.