After a number of phone conversations yesterday, I believe that the solution was found, largely thanks to both Brad Lauer and VexLover on CD. But to answer the previously stated question, yes, I did indeed have the latest version of the FTC Scoring software (as well as the IFI software).
You can indeed run multiple fields from the scoring table PC.
The issue we were having, however seems to stem from the UART to USB bridge drivers connecting the field to the PC. The problem with the Silicon Labs CP210x (CP2103) driver that we had was that the virtual com port driver "locks" to a single USB root hub, and since many computers now have separate root hubs for each USB port, when we tried to hook multiple fields onto multiple USB ports, the virtual UART com port driver couldn't decide which USB root hub to "lock" onto, and thus the conflict.
Now, from what I understand, there is probably some sort of software (or BIOS-related) solution out there where we could have bound multiple USB ports to a single root hub, but the easier solution was to simply plug a USB hub into one of the ports, and plug both fields into the hub. This way, both fields were on the same USB root hub, so the UART to USB bridge worked fine setting up multiple devices.
As a side note, I'd like to thank Harmonix for packing a USB hub with their Xbox 360 game
Rock Band, because that was the only one I own
.