OK, got it working.
Two of the pins on the RS232 connector (looks like RS232 anyway), going into the handheld portion of the unit were bent. The connector obviously had enough pins to provide power and some sense lines to the CANdi module to make it happy (flashing green light), but the bent (actually smashed after the connector was screwed together) were needed for successful communications.
The connector used is a cheapie (the pins are very soft). The good news is the soft pins allowed me to straighten both smashed ones with small forceps and not break them off. The bad news is this thing is never going to be disconnected again because they are not stiff enough to be sure they won't snag again in the future and bend/smash again. I suspect that cycle will not be very forgiving if it happened. That's OK, I will just keep the big cardboard box it came in as my case.