Stu,
I'm not sure that I have a good complete explanation for what you are seeing, although I have some observations and maybe a related explanation.
All of the Adler Barbour compressors made before about 1998, give or take, had a very high starting current of 17 amps at a nominal 12 volts. The "newer" ones are started gently by a more gradual application of voltage to the motor, rather than the "slam" start used in the older units. When the source impedance of the voltage source is very low, as it would be with well changed batteries that are well connected to the load, the starting transient only lasts for a short period of time ( 100 milliseconds +/- ??? ), and then the motor comes up to speed quickly enough that the high current doesn't last long enough for an analog or sampled digital ammeter to ever register the current spike. This is apparently just what you are seeing when you have a charging source connected to the batteries ( i.e. - the alternator or shore power ), and is in fact what you should generally see with batteries only. However, the fact that you see the weird, somewhat erratic current demonstrations when there are no charging source sources applied to the batteries makes me suspect that the compressor is seeing a badly sagging voltage at times.
Since low voltage will make a DC motor draw more current, I suspect that the long-enough-to-see ammeter reading at about 10 amps is when the motor is unable to get up to speed enough to cause the motor current to decrease to the lower correct value associated with higher/normal operating speed. When the 10 amp reading is maintained and the fridge continues to operate, the compressor motor continues to see a low voltage. When it goes to 10 amps, but then comes up to 5 amps, the compressor motor slowly managed to get up to "normal" operating speed, and the current reduces appropriately. When it goes to 10 amps and then shuts down, you probably had a voltage present that was low enough that the built-in low voltage shut-off of the compressor controller shut it down. All of this is, of course, long distance educated guessing.
The short answer is that I think you have a connection problem that is minimized or eliminated when you have either of your charging sources applied to the DC bus. What you should be looking for is a wiring configuration where the charge sources are connected ( in either the ground or 12 volt side, or even both ) electrically "closer" to the feed points to the fridge than the basic battery feed point connections. The problem you are looking for is between the batteries and the charge source connection points in the DC distribution paths. The way to get started is to connect your digital volt meter between the battery terminals and their respective path ends right at the compressor feed terminals on the compressor "pallet", and look for voltage drops across the 12 volt path, and then the ground path. Once you find an obviously large voltage drop, start moving your measurement point away from the battery end of the path and closer to the compressor. From what you are describing, I wouldn't be surprised if most of the drop is in only one feed line. The voltage drop should mostly go away by the time you reach the point where your charging sources connect along the paths. The internal fuse in the control unit or the fridge CB should be okay since they have to be down stream of the charge source connection points into the DC distribution busses.
Hope this gives you some place to start looking.
John