This is an old thread , but I have now exactly the same problem as the OP, and I expect more will have this issue as plugs get cheaper with higher ratings.
I am in 240V land and use 16/20Amp smart plug (Yes-chinese) so well rated within the demand of 12Amp…
though limited to the sample rate of the meter so not really expecting to see the inrush current from the cold heater, and resistance heaters that are installed don’t normally exhibit very high inrush… (I have not tested with an oscilliscope as mine is in store overseas, so can’t verify for my actual heater)
Anyway, once starting to try and control the boiler to optimise solar consumption (under load switching off, as opposed to prior I was just disabling ‘on’ when no daylight) the plug failed as per OP’s same symptoms. (Still switches but not once there is any load.)
I agree the solution is to go with a breaker with higher rating…
Anybody have experience with the smart breakers directly… i assume they still have a relay inside but with the rating of 60A I would assume much more capable for this application…
Tongou Amazon
Tuya
Moes
still chinese, but the schneider one takes my payback from a bit over 1year to nearly 6 years… 
I realise I will need to add another din rail box and to wire it in the circuit… but same if I add a breaker and smart energiser.