As for over-voltage, I just took a look at the schematic of your machine and it looks like it takes about 17-18 volts to shutdown.
That's a total shutdown until the over-voltage is removed, not the cycling reboot you're experiencing.
(One quick way to check this would be to repeat the car test with the engine running at 2000 rpm. The alternator would probably max out at ~14.6 volts.)
So, I think we're talking under-voltage at the machine input connector.
All this assumes your machine is working correctly.
It's either the battery or the voltage is dropping due to excess resistance in the cabling or connectors.
Like you said, I think you've cleared your cpap DC adapter cable (at least with your car battery).
As archangle suggested: you could test your battery and its wiring and connectors with a load. He also suggested twisting the plug in the socket to make sure you have a good connection.
OMMOHY addressed the battery voltages and charging very well.
That class of battery should work.
I guess what I've tried to address is the over-voltage possibility.
You could try ohming out you cable setup but most ohmmeters act pretty flaking below 1 ohm.
With the blower activated, you could measure the voltage going right into the machine by piercing the wire insulation with a couple of pins. (Seal up later with rtv and tape or heat shrink.)
Using the same technique, you can move up and down the cable or between a battery terminal and the load end, looking for a significant voltage drop (> 0.25V).
Hang in there, we will sort this out.