Charging would of course be possible with any case, but I think the original case is needed for pairing with new devices. But my wife has the same one, I could test it in the evening...

As far as I know, the case is paired with the headphones. I would contact the customer service and send in the headphones, if that is possible. Then the service should order the case and connect your buds. I recently had a Bud (defective) replaced by the customer service and everything was paired again...