I think you have a neutral - earth fault.
The neutral with no load is same voltage as earth, so if there is a low resistance between the two, nothing will trip, as no imbalance. But as the load increases the voltage between earth and neutral also increases, not by much, but enough to cause current to flow.
We use this
View attachment 306017 meter to test, and it puts 500 volt between the wires to be tested, it can destroy electronics, so care is required in its use, and the cheapest will cost around £35, it is always used when laying electric under floor heating, so the person laying it should have one.
I have seen it where a bit of damp toast in a toaster has caused a RCD to trip when kettle is used, as often we don't switch the neutral.