When BigCommerce orders are imported, FarApp will automatically change their status in the storefront to "Awaiting Shipment".
This process can't be modified to update the order to a different status; it will always be updated to "Awaiting Shipment". However, you could rename this status label in BigCommerce. For instance, you could rename the "Awaiting Shipment" label to "Sent To FarApp". In the API this status will still be known as "Awaiting Shipment" but in the BigCommerce interface it will be called "Sent To FarApp". This BigCommerce help document explains how to rename statuses: https://support.bigcommerce.com/s/article/Order-Statuses#rename