The "legs" of my playermodel are shown when looking down. I'm using the Unity First Person Controller template.
Character when in Scene view:
Character when looking down in Play mode:
You can use Layers to control what's visible to each camera.
First, create a layer for "FirstPersonHidden" objects, following the instructions in the docs.
For each object you want to hide, set its layer to this one using the Layer drop-down menu at the top-right of the Inspector.
Then, on your camera, open up the Culling Mask drop-down and ensure this layer is unchecked.
Now the marked objects will be hidden from your camera's view.