Interfacing to scale over RS-232
The solution turned out to be a couple of things, some of which mentioned by others in the comments.

Problems that needed to be fixed:


My level-shifter was a huge part of the problem. I switched to a MAX232-based chip and then I was a least able to get responses shorted lines 2 and 3 (Tx and Rx).
I needed a cross-over cable. I had tried this while troubleshooting, but because that alone didn't fix the problem, I had not idea whether or not that was the problem, or just one of the problems.
I needed to use the onboard UART (Serial1) rather than SoftwareSerial. The problem was that I needed to be able to set the protocol to 8 bits with no parity. This configuration parameter cannot be changed on SoftwareSerial, but is as simple as Serial1.begin(9600, SERIAL_8N2).


Thank you for the help, and hopefully my mistakes make the process easier for others.