US Navy NAVSAT of the 1970's used observed doppler shift to provide location information to vessels. The rate shape of the frequency shift curve (it would be an S from high to low like listening to a locomotive horn going past you) reveals the distance from the observer to the ground track of the satellite. (still have to choose which side you are on from it)

So the concept is sound. But if you can only see a small window straight up it isn't likely to work very well. Need the speed IN the line of sight to get a good shift.