Map arm movement (as plant) to lidar-measured distance (as setpoint) using PID
I think the two biggest concerns here are 'gain' and the bandwidth requirements of your system.
Without regulating the force applied to the lever arm from your actuator (bang bang) you will always be moving the arm with max acceleration, meaning you can't impose damping on your system through the control input, so it inherently has an enormously high gain.
You also likely have the issue of LiDAR limiting the bandwidth of your control loop; you can poll a position sensor orders of magnitude faster than the turnaround time for LiDAR frames.
I see in a comment that you are using a linear actuator, with some kind of CAN capable motor controller. Ditch the bang bang control. DC motor controllers are dirt simple. If your current one can't change the speed of your motor, you won't have difficulty finding one that can. Then instead of using time as your PID control effort, you can use motor voltage (or current if you buy a higher quality controller).
I also recommend ditching LiDAR in favor of an angular position sensor. I have spun out my own contactless magnetic angle sensors that are great for robotics (continuous angle output, retain the same angle across power cycles). You'll get higher bandwidth and resolution than with LiDAR. It looks like the linear actuator you're using also comes with encoder options, if you don't want to make your own.