Nigel is on a holiday, so I'll be following up on this ticket.
The way Toolset WordPress Archives work, they can't be loaded from within a different content builder like Elementor.
This means that you have two options:
1. You can either assign the Toolset's WordPress Archive to the WooCommerce archives and use only that for editing (without Elementor). The loop and the loop item content, both, will be handled by Toolset.
OR
2. If you'd prefer to use the Elementor template for designing these archives, you can continue using it and the loop part will be handled by the Elementor.
Inside the template, you can use the dynamic source option in the Elementor widgets to show the data from Toolset fields (as Nigel recommended) or use the Toolset's shortcodes directly:
In the WooCommerce loop within the Elementor environment, you can add a simple text/HTML widget and insert the required Toolset Types custom fields shortcode.