I think you didn’t exactly get my point about the projection matrix.
there are two problem that have to different solution. to make it simple you have to distinguish between a head and eye rotation
firstly check this video out : ://www.youtube.com/watch?time_continue=11&v=90kHhOUzeQc
you’ll see that without an actual change in the camera position you have what is projected to you changed. this is not quite equivalent of rotating the camera cause a game camera is not a complete equivalent of how our perception works. but in most cases you should be able to replicate it by just simply transforming the camera.