Sounds to me like a problem with the battery in the case rather than the antenna in the phone. I've had phones come in for repair making a similar noise when the battery is close to death. If the noise is gone without the case, then clearly the case is to blame, and I'm confident this is a battery issue rather than anything else, especially because you're using a Mophie (which I assume is a juice pack).
I would leave the case off, as this could develop into a fire hazard or worse. And contact Mophie about arranging a replacement; Lithium batteries that make noises are definitely dangerous.