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It does sound like a coolant temperature sensor issue which is being a bit over sensitive - as the engine temperature will increase slightly under load and this may be tipping everything over the edge and causing the ECU to go into safe mode due to the faulty sensor signal
It may also be worth replacing the thermostat
It may also be worth trying is a head gasket / block test kit as these take a coolant sample an analyse it for any signs of combustion gas - if present this would indicate the head gasket is on its way out