Note at, with GitLab 15.5 (October 2022), you now have:
API endpoint to get project transfer locations
We added a new Projects API endpoint that returns a list of groups to which you can transfer the specified project. Previously, you could select which group to transfer a project to from the dropdown list, which didn’t include groups made available through group sharing. Now, the new endpoint ensures that these groups are included too.
See Documentation and Issue.
That can help if you have to script moving a project to a group, and need to propose a list of possible targets.