After ruling out SCSI pass-through and iSCSI, I found a link referencing PCI pass-through on ESX servers. This requires a system that supports Intel VT-d and essentially ties a PCI card to a running virtual machine. The VMware feature is called VMDirectPath I/O, and information on configuring it can be found at http://kb.vmware.com/selfservice/microsites/search.do?language=en_US&cmd=displayKC&externalId=1010789
Once I found a host that supported VT-d, I was surprised how easy the setup process was. Just configure the VMDirectPath device, reboot, and add it to a VM. Using a QLogic HBA, my Backup Exec server easily discovered my tape library and drives.
Performance for this setup was great! I actually edged out my prior configuration, but that was likely due to the hardware I was running on.