You have to target Woo notifications by referencing their notification methods. Basically there isn’t a “catch all” - or I haven’t found one yet. You will have to notice a “notification” that doesn’t get picked up by “Admin_Notices Snippet” and then add it to your list.
Here is the class that contains all of the WC Admin Notices. WooCommerce Code Reference
ob_start/clean() converts the output into a string. Simply wrap the call arround the
Return info from Child Sites