Microsoft Support and I were unable to load the driver manually when booting into WINPE. They suggested two other options to get around the problem.
Option 1, Creating a customized WinPE image
This would require injecting the correct network drivers into the WinPe image and boot the server from the customized image. The idea is that the correct NIC driver is loaded and we would gain access to the network and allow us the read the file over the network. See
http://technet.microsoft.com/en-us/library/cc709665.aspx .
Option 2, Use an USB drive to read the backup
With this solution the backup file is copied to the USB drive. The server is booted with WinPe. There is an UI that allows you to load drivers for remote storage devices and read files from the device.
We decided to go with option #2.