The default output is comprised of a series of action hooks. You can override the defaults by hooking with a priority arg higher than the one you need to override. A value > 10 should do it in most cases.
Your callbacks would need to keep track of how many times they’ve been called in order to only act on the 7th item. Keep this count in a global variable since local vars are destroyed when your callback returns. Refer to /woocommerce/templates/content-product.php for the action hooks used.
Instead of dealing with individual hooks, you could override the entire template file. Check the comment header of the above referenced file.
If you need more specific guidance beyond the above, I recommend asking in the WooCommerce dedicated support forum.
