I have some new information.  I tried different settings of dial C, the
gas level, from min to max.  While doing this, I discovered I can keep the
flame going by continuing to pump on the air pump (A) while holding the
by-pass (B) in.  Then, doing that repeatedly, seeing what would cause it
to shut off, I found that I could release B and it would stay light.  When
I stopped pumping, the rumble of the flame would diminish, so I would pump
more and it would come back.

The odd thing (or, more odd) is that a couple of times the flame acted
like normal, it started and I could see the flame in the window, and the
LED on the panel lit up.  This would happen within a few seconds of the
normal flame.  Then, the flame went away and I heard the rumble.

Is the flame burning in the wrong place?  Burning inside the exhaust pipe
instead of down in the burn box?  I tried playing with the level (C) to
see if perhaps there is too much gas pressure, and it's preventing enough
oxygen from getting in.  I couldn't find a level that would keep it going
in the normal flame appearance mode.  It rumbles in the pipe as long as I
keep pumping.  When I stop pumping, it will rumble on for a minute or two,
then there's a click and it goes off (or it goes off first... seems
simultaneous).

Maybe the thermostat is okay and the flame is in the wrong place.  When I
took the burn box back together, I put the pipe that comes off the gas
nozzle so that the slots in it are up (when the fridge is standing
upright).  I assume the flame burns in those slots, where the gas going
through the pipe meets the air in the box.

I have built a manometer and checked the pressure out of the propane tank
regulator and set it correctly.

--
==Leonard