Possible to measure voltage while charging?
The circuit is wrong and doesn't represent a real world charger. A charger, even if modeled as a voltage source, will have a series resistance so when you measure the voltage across the capacitor you see its voltage and not that of the voltage source.

Most real world battery chargers have several operating modes, one typical mode of charging is that they operate at constant current until the battery reaches its nominal voltage and then at constant voltage until the battery reaches full capacity.