It seems to me the problem is transferring a file before it’s complete, not that you’re deleting it.
If this is Linux, it’s possible for a file to be open by process A and process B can unlink the file. There’s no error, but of course A is wasting its time. Therefore, the fact that rsync deletes the source file is not a problem.
The problem is rsync deletes the source file only after it’s copied, and if it’s still being written to disk you’ll have a partial file.
How about this: Mount mass as a remote file system (NFS would work) in speed. Then just web-crawl the files directly.