Dang close on the answer, but here is the deal
The thermal timer controls the cold start valve.
Next to it is a coolant temperature sensor for the ECU.
There is a bad deal on the ECU coolant sensors that can make what you are seeing.
In the 90s BMW did a coolant containment modification.
They added some valves to the heater and the throttle body heater lines.
The used the wires feeding the ECU coolant temperature sensor to power the added valves. So as the valve get older they will affect the sensor and cause exactly what you are seeing.
A specific temperature when the engine will go full rich and cause the engine to sumble, blow black soot, and stall.
Full cold it is alright.
If you can get it past the bad temperature it will also run alright.
I have seen three do this and only figured the fix by luck.
The added valves have a connnector the same as a 30 amp green bladed fuse so unpluging the valve and inserting the 30 amp fuse will bypass the issue.
Later,