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.