Having now used the 100V/100W outlet for the cool-box via the 'Chinese electronics' I can report.
The transformer works perfectly, there is no switch for the outlet, but it is 'ON' all the time the engine is actually running, dropping out when the engine stops.
It has not enough 'welly' for two transformers, and just goes 'OFF' on overload, resetting itself after removing the overload situation and restarting the engine.