The third-party program is something obscure that you probably haven't heard of - its 2D optical code reader software from Micronic/Steribar. In their software, I can set the name of the Virtual Serial Port.
Anyway, I have been trying different things here. I tried upgrading to MAX 3.1.1 and Serial 1.7, to no avail. However, I had another idea. MAX shows "COM3", which does not correspond to any physical port, so I tried naming the Virtual Serial Port to be "COM3". This works! I can read the 2D optical codes in LabVIEW by simply selecting COM3. Going back to HyperTerminal, that also works, but it shows "COM3" twice in its port selector list, and both selections communicate with the 2D optical code reader. MAX only shows one listing for COM3, and using it to try to communicate with the code reader is unsuccessful.
Well, it works now with LabVIEW, so I won't touch it anymore.