Got home the evening of the 9th, opened the laptop & received a message basically saying "I'm broke so bad you've got to take me to the repair shop". Picked it, (and the broken HDD), up this morning - on the way home from the hospital where my surgery had been canceled. (grumble) Got up early, skipped breakfast, drove 1 1/2 hours, only to be sent home 'cause the doc was "busy with a trauma patient" (/grumble)
Had a chat with Nick1911 where we determined that what's most likely happening is there's not enough air coming out of the floor registers to supply the air demands of the heater. A couple possible reasons are that the supply ducting is way too small (my idea) or there's a leak in the supply ducting (Nick's idea, most likely the correct answer).
DW talked with the AC installer, his idea was that there's a problem with the chimney - that the chimney was blocked, not letting the exhaust out and it would backup into the room. He recommended a chimney sweep, she called and made the appointment. I talked with the sweep when he called to confirm, he didn't seem to hear me when I mentioned that when the heater's running you can feel the cold air coming out of the fireplace even when the fire's out, the damper's closed and the doors are shut.
The chimney is now clean, and we're still having the same problem.
The sweeps' suggestions were to a) don't use the fireplace b) open a hole in the wall to feed air into the room and/or c) open a window near the air return.
Inspecting the ducting would be a serious pain in the ass. It runs under the house, with about 6" of fiberglass insulation held in place by underbelly wrap (basically a tarp) covering the entire underside of the house. I am not in any shape for tearing all that apart.
Next step will be calling around to see if there's an HVAC company around who can inspect the ducting, preferably with a camera.
Today we just shut off the heater & used the fireplace, we'll reverse that tonight.
ETA: I did a quick & dirty test on the air flow from the floor registers. Turned on the house fan and held a few squares of TP over each register. The house is a rectangle, there are two supply lines running lengthwise, one on either side of center-line of the house. The registers on the rear side of center-line are definitely putting out more air than the others.