For a short period you could run without a thermostat, but the engine will be circulating fluid continuously, and the engine will not warm up to operating temperature. If your car is running hot, and the radiator has fluid, your thermostat is probably sticking and needs replacement. Better to have the car run "cold" with fluid running through the engine continuously than to overheat the engine and have it hot. I would get the thermostat replaced. Not normally expensive, but as it is a BMW prepare for the worst!
Bel Air Car Guy