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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.
If you look at some mid 50s catalogues and search for 8 speed bicycles those rod actuated front changers normally used 46/49 3/32" chain rings combined with a 4-speed rear (which would have been 3/32" as well). For example the 1955 Dawes Emblem 8 specifically mentions a 3/32" chain is fitted! Same for the 1956 Lenton Grand Prix, also an 8-speed with 3/32" chain.
I would be tempted to use a 3/32" freewheel, maybe even a 5-speed.