Hello Adam Czajczyk ,
Thanks for getting back so quickly! I'm thinking maybe something in the Integrations Area. A WooCommerce one. Once activated, WP Smush would Resize the images to the chosen sizes in the WooCommerce settings (or maybe even an override in Smush settings). While resizing, you can have an option of making the images (or thumbnails or both) into a square. When that option is activated, you can choose the BG color or the option of choosing the BG based on category, then on each category, you can select the color you want. It would be nice to have a default one and then we can override it in the category pages if we want.
I'm not sure if this is what you were looking for, but I'm open to further discussions and testing/providing feedback.