I've now added the two FHRS IDs to the Playhouse itself (the theatre).

This just leaves the question what to do with the FHRS ID for "Pizza al Teatro / Little Picardy": we don't currently have Pizza Al Teatro mapped.

My feeling is Pizza Al Teatro and The Boards are the same place, and share an FHRS ID with Little Picardy. It should be easy to confirm this by going in and checking the menu.