The Manufacturer SKU is used to group products together but, unlike WooCommerce there is no SKU on a parent product, so it will not appear in the product normally. It is possible to map this to something but you would need to "make room" for it.

The barcode field can be used but this is really meant for the GTIN or similar identifier or the Retail Express product ID. It is not wellsuited to being a tag as it does not re-occur and will add a lot of new tags.