Wear a safety harness... that may work if you are going slow in a straight line, but I would worry about inertia throwing the camera side to side when you take turns. And you still have to take some of the vibrations that would run from the base, through the arm, and into the camera, perhaps using your arms to absorb some of it. If you want a handheld-but-smoother look, something like an EasyRig maybe, or an arm with a bungy cord, again with you and everything safety harnessed. At minimal, you could mount the camera to a hi-hat or short legs but use an vibration isolator under the head.
The main problem with not using a camera car is not having that forward platform for shots chasing the bikes. If you don't need to operate the shots, perhaps you can mount the camera to the nose of the car using something similar to a hostess tray or with speed rail. With a beefier speed rail rig, maybe you can even put a vibration isolator and then a geared head with Hot Wheels so you can operate the shot. At some point, however, it may be simpler to just get a camera car so you can operate the camera on the nose.
So whatever you can to make the ride smoother, smoother road, take some air out of the tires maybe, etc.



