No, the batteries from the Power Kit products do not contain the same contacts that are present on the batteries used in the Modular Power Station.
There is a possibility that the battery might not be resting correctly inside the Power Station. Try opening the cover and adjusting the battery. If the issue persists, please contact customer support. Our customer support can be contacted here.
If the Power Station is tilting forward, try checking to make sure the switch on the back of the Power Station is set to the “locked” position to secure the unit to the console.
No, Modular Power Station is not compatible with Xbox One S.