I think I found the issue and why the camera is moving. The camera is moving when the collision shape moves because the collision shape enters the ground, causing the entire KinematicBody node to be moved upwards when the player moves while crouching. I believe this is what is causing the issues you are experiencing.
Here is a GIF showing what is going on, where the green sphere is the position of the camera, and the red cube is the position of the floor when the player is not crouching:

There are several ways you can go about fixing this. Changing the size of the capsule CollisionShape node that is used for the player is one way, but personally I think of the easiest and most flexible ways is to have multiple CollisionShape nodes that you enable and disable as needed.
I made the changes to the project to use multiple CollisionShape nodes and now the camera does not move with the collision shape. I have uploaded the project to my Google Drive if you want to see the changes I made.
Hopefully this helps!