If power was really the problem I'd just use a battery inside the cage.
Related, this is a neat experiment which powers a galvanically isolated circuit optically using LEDs and a solar cell: https://youtu.be/9JinSfCKuNQ?t=823&si=kBwesUsVgAGBVVwq
He didn't need to block RF though, the interface seems mechanically challenging to build into a cage.