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I think a narrower chain will work, the chain doesn't engage with the smooth pulley wheels, so it shouldn't bind on those, the pulley plates will be a little further apart than a 3/32 RD but it'll probably work. It probably wouldn't the other way around.
I have a 4 speed Simplex 51 (model after TdF) - but that is marked 2.38 -so not a 1/8" chain - but I think they still made them in both sizes.
Yes, I see what you mean. The thing is my Simplex Tour De France rear changer is marked for a 1/8" chain, so I assume it is set up and spaced for such, although of course some 50's style bodging could be in order to get it working with a 3/32"....