Is there any way of estimating the flow through a screen in situ, such as a propeller, as a test to see whether the water level can be maintained?

There are borehole flow meters of various types that can be used to identify the ambient flow through the screen zone. There are magnetic conductance flow meters and also heat-pulse flow meters. They can pretty accurately measure the ambient flow of water through the well bore, and that should give you some idea of whether or not you have significant ambient flow which is helpful because if you do, your ability to get stabilized conditions is greatly increased in a shorter period of time.