This issue is similar to another posted here , but in your case you need a 3rd camera to render just the FPS mesh(weapon/cockpit I assume?)
With a setup like this you should follow this convention:
Base scene camera:
Normal camera, rendering all your scene minus the UI and FPS layer
No Post Process layer
FPS/UI camera:
layer mask set to FPS layer and UI
Post Process Layer
Higher Depth than base camera
Also both cameras should have the same HDR/MSAA settings to stack properly.
If you want the UI on a separate 3rd camera(I suggest against as the additional cost of a 3rd camera is not worth it and post doesn’t affect it by default) check the other post I linked to and then treat the FPS camera as the scene camera in that setup.
Let me know if you have any issues setting it up.