Add a tag to a product using a button press or checkbox

I’ve been working on this solution for a few weeks and have not figured it out yet. I am trying to set up a page where I can add a tag (or some other field that I can filter by) to a product using a button click, a check box, radio button, or something similar.

Ideally, I would use a WooCommerce product table to display a list of products, without an option to add them to cart. Instead, there would be a column in the table that contains this button which, when pressed, adds a specific tag to the product in that row. That would allow me to create another table which filters all the products in my store by this tag.

I have tried using Custom Post Types to create this tag, and then represent the presence/absence of the tag per product using a checkbox in a product table, but that has not been working.

If anyone has any suggestions for how to achieve this please let me know!