As it turns out, we actually cannot share the UPS thru ShipStation account's shipper number. This is simply a limitation of that UPS thru ShipStation account, platform wide.
So, to utilize the type of shipping you described, you would want to have your own direct UPS account and share that account's shipper number.
The reason the account cannot be shared with third parties is that the Terms and Conditions for ShipStation's UPS offering expressly prohibit doing so without your securing prior written consent from UPS. Here is a copy of the Ts and Cs that are signed when granted access to the UPS from ShipStation integration for your review.