Hi Lars,
The Fusion connector is labeled as "Google Team Drive." When I first started exploring Google Drive integration, it took me a minute to realize that THIS connector controlled the Google Shared Drive, as you state.
This connector does not offer a way to move a folder on a Shared Drive.
I contacted Adobe Support, which confirmed that the HTTP -> Make a request module would be required. I have contacted my IT team to see if someone could help me build the correct API call, which is beyond my capabilities.