Retrieve deleted S3 files?

No, it isn't possible. In AWS, you are in charge of your data: when you delete the last (or only) copy, AWS won't have any hidden copies of the data you decided to delete. What's deleted is gone.

AWS Backup for S3 would've been the way to instruct AWS to make a copy of your data outside your account and bucket, but it won't do any good after the data is already deleted.

Versioning alone would have caused the objects only to be "marked as deleted", rather than actually deleting them, but even then, with sufficient permissions, you could've deleted the earlier versions just as easily as the only version. You would need to have used S3 Object Lock on top of versioning to prevent some or all versions from being deleted with any permissions.