If it's an angular tilt you want to measure, then the "roll" angle will provide helpful information.
But from your description, it sounds like you want to measure lateral left/right (X-axis) motion; from my experience the best solution is to mount a fixed wheel (omniwheels work well for this) with an encoder on it that measures the linear motion in the X axis.
As you've mentioned, the accelerometers on todays inexpensive IMUs are not accurate enough, so I think this type of passive "follower" wheel is the simplest thing that could possibly work.