If the fan does then turn off, your best bet would be to go to a air con agent(not bmw£££).
I do not know what type of air con system you have on your car ie orifice or expansion, but each have a temperature sensing arrangement on them.
Air con can be a mine field and one that I am not fully conversant with, so I can only advise a air con person if the fan does top with fuse pulled, unless you want to do without your air con, which I am sure you don't, as it is a good thing to have.
tabetha