Steve, when operating the dipping lever the whole case including mirror 'dips'. The mirror is attached to the case via the lips ( and in my case the black trim also). When you slide the dipping lever across it makes the case move in relation to the stem plates which remain stationary. There is no contact of the mirror with the dipping assembly.
I see no reason for not being able to replace a 'black lever', the black levers I've seen are just heat shrink tube around the metal dipping lever. If using a white plastic ball it would require a rectangular slot to be cut into it so it can be pushed on. Perhaps a small drill could be used to produce 3 holes in-line deep enough to keep the white ball on by pushing it on.
Dont know of anybody who does re-silvering in UK but havent searched to much.
Suggest you try antique restorers like this one
Taylor's Glass
Also having looked at various photos there are no black trims in any original GT40's I've seen (but does the one pictured below have one???, I have also seen early GT40's with a different stem and I think is located to the roof or attached to the glass near the roof.
Its also surprising how far offset to the left (for right-hand driving position) the mirror is on some cars. I assume you get better views, see photo.
Andy