Heck, why does it have to physically move at all? Why doesn't this effectively come down to a rename within AWS' system, like when you "move" a file within the same hard drive?
I've seen many similar situations that boil down to "UPDATE SET x = y WHERE z" require support tickets at a minimum, or are flat impossible because even their internal staff don't know how to do it.
We ended up writing a Go program to copy the objects from one bucket to another to help with the parallelization of migration. This was also before AWS had announced S3 Batch operations so I'm not sure how much better it would be to use that today. The deletion of the bucket also took us over a week. Even though we had deleted all objects in the bucket due to the eventual consistency nature of S3 we weren't allowed to delete the bucket until all objects were fully removed from S3. All we could get from AWS support was to wait a few more days and reach back if we couldn't delete it then.
Edit: depending on your object naming scheme you might also run into the S3 prefix rate limits.
If the copy Api supported wildcards there wouldn’t be any discussion at all on this