As your application does not need on-the-fly sync between the devices, running an SW-based time lag estimation is also one way to do it.
The following materials might help to start with.
Spatiotemporal camera-LiDAR calibration: A targetless and structureless approach, C Park, P Moghadam, S Kim, S Sridharan, C Fookes
However, upon basic online search, I found that we'll need a frame synchronized rig for such purpose
This is not a simple problem and there exist several solutions, based on project needs, to achieve what you are trying to do. A basic search isn't enough. This forum is for specific questions / errors you want resolved.
As a starting point, you can take a ...
Go back and look at the camera idea again, but this time put the camera at the end of the "tube" (where you sketched the lidar) looking along the tube just like the lidar does. It will see everything the lidar would, except better. Cameras already have a dense, high performance array of sensors (at the image plane).
A simple webcam would meet your ...