Where are you that electricians are so hard to come by?
Can you get your hands on a clip-on amp-meter from ebay if nothing else. This would enable you to check the current drawn by the heater. It will require access to one of the individual Line or Neutral wires inside the cable or local isolating switch (beware of the little men running up your arm). A 3.4kW heater should be drawing around 14.2 Amps but if the supply voltage is nearer 230v than 240v which would be the basis for an older model it will be lower around 13.6 If the heater is the culprit a higher current would be passing. The type of fuse or circuit breaker at the supply will influence how much current can actually pass without operating (blowing) the fuse or tripping the circuit breaker. An old rewirable fuse rated at 15Amps will pass close on 30Amps for hours without operating so you cannot rely on the fuse recognising a defect. I'm still suspicious of a branch off the circuit which you do not know about. Checking the current passing out of the fusebox to the heater and comparing it with the current arriving at the heater would prove this one way or another. You could also monitor the current in the meter tails as you switch loads on or off although for off peak this will involve going to bed late or getting up early