Answer
The Staged Billing Profile screen is actually built on the CustomerOrderLine Logical Unit under the hood — it isn't its own independent LU — so any custom field added there is really a CustomerOrderLine custom field, and the Staged Billing Profile window itself functions purely as a read-only data-representation view rather than a genuine data-entry screen for that LU. To populate the custom field's value, enter it from an actual editable screen tied to CustomerOrderLine instead, such as Customer Order > Order Lines tab or the Customer Order Lines table window — once entered there, the value correctly shows up (read-only) on Staged Billing Profile as well.