I did something similar in my app here to make the camera orbit around the mountain and stay pointing towards it: Projecting flags on Matterhorn mountain - behind the scenes
I did it by having an invisible entity on top of the mountain that the camera is tracking. It sounds like this is basically what you’d need, your entity is moving, the camera is changing direction to look at it but not position. The source code for that app is available in that forum thread if it helps.
I think ideally we could bring this into CesiumJS as a simple camera.lookAt(entity.position) and you’d run that every frame to turn the camera to the moving entity. We’d be happy to review that pull request if this is something you’re working on!