If you want to take the middle server out of the loop, you'll need to get one of the other two servers to initiate the copy. A combination of PsExec (as suggested by moshen) plus RoboCopy would do the trick efficiently and quickly.
RoboCopy has a bunch of command-line switches that make this kind of sync easy to do - just be sure to test thoroughly before you start mirroring deletes (
/MIR), else you can purge both directories in the blink of an eye!
Depending on your network infrastructure, you might be interested in the
/IPG:n switch, as it forces RoboCopy to wait between packets, allowing other network traffic to get through.