I guess I'm still not sure if this falls into the "leaky bandage" category or if it qualifies as a legitimate solution, but I did find a way from Microsoft that works for me to turn off automatic detection by Windows of devices connected to the COM ports. As with many similar articles, this one cites a talkative GPS device as a typical example of why this sort of thing is needed sometimes.

The referenced page also notes that--as Jan suggested--the situation can get even more complicated if a USB-to-serial adapter is involved, because it might incorrectly detect and try to emulate a serial mouse instead of a serial port. Since that's beyond the control of Windows, the Microsoft fix won't work and you would then be trying to sort out problems with the adapter along with everything else.

-- Russ