The Woo Custom Stock Status plugin stores stock statuses separately from WooCommerce’s default stock status values. Most product feed plugins only include standard WooCommerce fields (instock, outofstock, onbackorder) in their feeds by default.
To include custom stock statuses in your product feed, you will need to check whether your feed plugin supports custom fields/meta fields and then add the Woo Custom Stock Status field to the feed mapping. The exact method depends on the product feed plugin you are using.
Please let us know which product feed plugin you are using, and we can advise whether our custom stock status can be included in the feed and how to configure it.