An assist dog of any kind would of course be allowed on the train, but I think I'm understanding your question to mean that this is a pet and not a helper or service animal, right?
The important thing to do is contact the train management yourself. If your dog is small and could fit inside a reasonably sized travel crate, that might be the answer. Airlines allow pets in the cabin this way, under certain conditions. Of course you'd have to plan your connections so that your dog has ample opportunities for "relief" stops, which might be difficult to do when traveling by train.
Good luck!
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