It’s common to find theme that you can’t simply override certain file by mirroring the file structure in your child theme.
In this case the cart.min.js is most likely included via wp_queue_script so you gotta find the handle of the script by looking at the source code of WooCommerce (just search by wp_enqueue_script). Then you can run wp_dequeue_script on that handle.
Lastly, you run wp_enqueue_script on your own js file to include it in your template.