the proper way to solve that is to press the stage button, and then you can turn on the engine on the 2nd stage which has its own fuel. and when that runs out of fuel, then press the refuel button.
alternatively, you could just press the refuel button without stage separation, which would refuel the 1st stage.
refueling while suborbital is allowed, but that's unrealistic. for realism, refueling should only be done while in a stable orbit, such as with a periapsis of more than 300km (for earth). however, launching from large planets currently requires unrealistic measures.