DoggyCam… hadn’t thought of that one. Not really the effect I’m going for, but a good suggestion.
How about a single track down the middle of the rink, with the camera mounted on a jib? As the singer moves clockwise around the rink, the dolly moves back and forth with the jib swinging clockwise to follow him around the curves. Would require some heavy gear and talented operators, as you said earlier, but this may be the simplest way to do it.