I've seen that "look" before John. Not on my H2 but when these cameras are attaches to Quad Copters such as the DJI Phantom. Do a search on YouTube and be amazed!!! I so want a Phantom now!!!
Anyhow, 4 spinning blades that may be out of balance is extreme example of the Jello effect. I've never, ever seen it that bad with the camera mounted in a car. With the copters they say one of the best ways to solve the issue is to change to 60fps rate. Thus closely matching the inherent vibration of the copters blades. But you're not in something with spinning blades at close to 10K RPMs. The oscillations of a car are not that rhythm and that electronic sound is very, very odd. Seems like something is coming apart.
I have shot many clips from either inside my car or outside mounted on the body. I also just recently mounted my camera inside a Jeep that was going off roading over some heavily rutted dirt roads and big bumps. All the footage was spot on perfect. This was for a Nationally televised cable TV show so the footage needed to work. For that I used my Panavise suction cup mount with the GoPro tripod mount attached to the camera. No Jello footage at all. The other GoPro camera was mounted under the front end of the Jeep to capture the tire going over the road and obstacles. It was hard mounted to a sticky and armed out just a bit. If any of the cameras were going to shake, it would have been that one. Again, spot one images with no Jello effect and none of that electronic noise.
One suggestion I have is to get a thin layer of rubber to fit between the bar and your mount. It might help isolate the camera better, but is will also give the mount more grip as it bites down on the bar.
I'm with the others, do more wide and medium and narrow camera tests just to see if that's part of the problem. And if you can get another H2 to test your against the other camera. If you can. I have recently seen the price of the Hero 2 drop a lot. The H2 Outdoor kit, with the WiFi bacpac can now be had for $199. I might pick up another one.