Your LIDAR will need some active stabilization. Under gravity alone it will sway like a pendulum.
You could start with a web search for "servo gimbal". Resulting hits show complete devices, arduino projects, tuning strategies, hobbyist guides, etc., for drones, copters, UAVs, etc. Your LIDAR mount might benefit from work done for mounting cameras. That suggests a look at camera pan-tilt setups, which sort of combine gimbal and servo motors.
The solution, whether manufactured or DIY, will have a gimbal mount, an accelerometer, servo motors, and a microcontroller.
The gimbal is like a 2 degree of freedom mounting bracket. Your LIDAR would be rigidly fastened to the moving part of the gimbal, and the gimbal's base would be rigidly fastened to your airframe.
The accelerometer, mounted to the LIDAR, measures the tilt (i.e. 1 g, pointing down). It'll be only approximate if your airframe is being chaotically accelerated by violent wind gusts, or in freefall!
The two servo motors tilt the gimbal in the two degrees of freedom. Model airplane servos conveniently bundle motor, geartrain, and control electronics in one little package.
The microcontroller provides the smarts to tie it all together. Its basic control algorithm is pretty simple. Continuously, it reads the "g" vector from the accelerometer, and commands the servos to move the gimbal (hence LIDAR) to counteract any tilt. Fancier coding can try to deal with anomalous responses, or respond to commands from the transmitter.
If you go the DIY route, the arduino is popular and has a broad, supportive user community, but it is by no means the only option (or necessarily the best). Accelerometers are pretty cheap and light. Servos are engineered to be light.