ShippingEasy Integration Guide
If you're using ShippingEasy, you should follow this guide to sync BYOB orders to this fulfillment app.
By default, ShippingEasy will automatically sync any Shopify order with an "Unfulfilled" status. Any add-ons or adjustments made after the initial purchase will not be synced with the current settings.
However, you could turn off the automatic downloads for that store by going to Settings > Stores and Orders > Edit Store Settings > Orders tab. This would mean you would need to manually click the sync button of the Orders page in order for orders to pull over.
Reference: Syncing Shopify Orders