How can I change the Fulfilment Status for multiple Item Fulfilment Records?

I have a Saved Search that lists them but I can't edit the Status using the Inline Edit Option.

I have looked at a Mass Update, but this doesn't appear to have the option to update the Status field.

The 'Fulfil Orders' page for mass fulfilling seems to be based on Sales Orders that have not yet been fulfilled.

All of our orders are fulfilled to 'Packed' in the warehouse, and then our 3rd party Delivery system should mark them 'Shipped' when the delivery is complete.

However, there can be some orders that are delivered outside of the 3rd party system, and these need to be marked as Shipped manually. We are looking for a way to do these in bulk rather than individually.