As long as you have coolant in the heater it adds to engine cooling when on full with the blower fan as you are taking more heat out the system.
In old classic Mini's this was always a way of getting excess engine temperature down in traffic even on a summers day...as your passengers roasted!
However if your heater ever goes cold while set on hot stop driving as it means there is no coolant in the heater matrix so there is a leak!
The cap's pressure release valve may be chipped and faulty/broken from new and therefore not maintaining pressurised system....this would cause overheating and as a bonus be a cheap and easy fix.......fingers crossed!
I have posted more info here which may help: Solving MINI Cooling Problems.......... - Mini Cooper Register Forum Quotes from the NAM thread:
"The cooling fan first stage speed is like a base point to provide a constant regulation of temperature while 2nd stage fan is for peak temperatures reached when car is stationery or under load, it quickly brings the temperature back to the regulation zone.
At 105C the low speed should be on
At 112C the low and high speed should be on together
OR if the refrigerant pressure from the A/C compressor is good and Air Con is on then the fan should also be on."
"In a perfect Mini cooling system, the low-speed fan will come on (and stay on) when you turn on the A/Con - always. If the fan does not come on (low-speed) when you turn on the A/C, but then comes on at high-speed after a few minutes (you'll know it's on high - very loud), those are the symptoms of a burned-out resistor."
"I decided to spend some time observing the fan in my garage and after a while of having the car on and the fan still, I decided to turn on the A/C and the fan did turn on. I turned the A/C off and the fan remained on. Waited some time and it stayed on at the same speed. So I got back in the car, turned the A/C back on and rev'd it for a while and then... high speed came in.
So in short, I was able to determine that my fan does indeed work at low speed and high speed."
Your fan sounds like it is faulty.....I think there are several similar 'fan problen threads on Mini2 if you search or also try the USA Mini forum NAM for useful info: 1st Generation MINIs - North American Motoring