Hi Steven,
I'm afraid JetS3t's renameObject() convenience method dates from the days before S3 supported 5GB+ object sizes and it is unable to copy large objects. Behind the scenes, this method just performs a standard PUT with the "x-amz-copy-source" header.
To copy large objects you will need to use the S3Service#multipartUploadPartCopy method, along with all the other fiddly overhead of initiating a multipart upload in advance and completing it when all the upload/copy operations are completed. JetS3t does not currently have a convenience method for handling all of this work, and even though you're notionally doing something very simple when just "renaming" a large object the API steps are actually fairly complex.
See JetS3t's code for the MultipartUtils#uploadObjects and S3Service#putObjectMaybeAsMultipart methods which are convenience methods for uploading multiple files as multipart objects, or normal ones, based on file sizes. You might need something similar in your app to handle object renames, using a simple approach when possible or the multipart-upload-copy approach when necessary.
Hope this helps,
James