Unfortunately, Shopify doesn't allow us to set the fulfillment date when creating orders.  The fulfillment date you see in your Shopify admin is the created_at or updated_at field, which is populated by Shopify and a read-only field to third-party apps.

When importing/creating orders, Shopify allows us to set the process_at date on the order and the transaction, but this field does not exist on the fulfillment.

Shopify controls the fulfillment workflow, so when importing/creating orders, we can't create the fulfillment; we can only update tracking information or mark fulfillment orders fulfilled without providing any date fields.

Here is a forum post from a Shopify employee that gives more context on why we cannot set the fulfillment date when importing/creating orders.