Craft 3 - Moving assets between folders causes apache connection to max out
I would remove the overhead of the CMS all together by using an app like Filezilla, Cyberduck or similar app to move files. There are plenty of tutorials out there on how to connect to an S3 bucket using one of these apps (eg. https://stackoverflow.com/questions/45293159/how-to-setup-filezilla-pro-to-access-to-my-s3-bucket)

After you do this you will most likely need to update your asset indexes in the control panel:

Utilities → Asset Indexes (or Settings → Update Asset Indexes if you're on Craft 2)