Hi Sowerberry,
Welcome here, I don't know anything about the Cox setup but if you have the chain fitted where you think it should go then that will be the way it has to go, then there will be an adjuster somewhere to slacken off the chain. If the adjuster won't allow any more adjustment then I'm guessing the PO fitted a chain that is too short or if as you say it is riding up on the pulley then the chain might be the wrong pitch or too narrow. They could have tried to fit a motorcycle chain and that is why they gave up
