Do the test I suggested with a headlamp bulb, and measure the drop from the actual battery terminals to the actual pins of the bulb - check both earth and positive, to see where you're losing most. Switches, fuses (and particularly thermal breakers) are places you make losses, as well as thin cables.
I used the pump as the test load after adding water to the fresh water tank so I didn't have to run the pump dry.
Most of the 12V loads (water pump, lights, radio, gas frig, gas water heater, USB charging ports, 12V outlets) are all returned to a ground bus bar (ground mecca) that sits behind the fuse panel. Measuring between the mecca and the 12V input to the fuse panel, I noticed that this voltage dropped (0.1 to 0.2V) more than I expected even as low amperage loads (like LED lighting) were turned on. When I turned on the tap to start the pump, the voltage dropped from 12.7Vdc to ~11.9V.
As suggested by Nigel, I ran a test wire into the trailer from the negative pole of the battery (externally mounted on the trailer tongue) and measured between it and the ground mecca while running the pump. Viola, the difference accounted for most of the 0.8V drop . Next I measured (pump running) between battery - and the trailer frame; less than 5mV. Then I measured between trailer frame and the internal ground mecca; seeing the 0.8V again...
Carefully tracing the connections between the trailer frame and the internal ground mecca, I found the problem. The trailer builder had not connected the #8awg wire that runs through the floor of the trailer to the ground mecca. It was connected at the frame end, but the internal end was just bundled with some other wires, and had never been stripped and clamped under one of the screws in the ground mecca...
After connecting it, the voltage drop when the pump starts is now less than 0.1V. I substituted an old, partially discharged automotive starting battery for the new battery that came with the trailer, and now the radio does not reset even with the system voltage at ~11.5V.
This was a maker's oversight; a quality control issue at the trailer maker's factory. Its manifestation was quite insidious, because there must have been a secondary ground somewhere, likely through an appliance, which has small gauge wires leading to it, causing the large voltage drop... but yet most of the systems worked well enough that we had spend a few nights in the trailer before I found and fixed the underlying issue.