The car to car ramps are right and left only in line with automobile wheel tracks. Since there's no center ramp there is no way to load a Spyder directly into the auto car. Motorcycles are loaded onto flat cars, or wagons, with a wheel track that fits the ramps. The flat cars are equipped with front wheel chocks that are too narrow for the Spyder rear wheel. The Spyder would have to be loaded backwards and even then the fender, mud flap, and hitch if it's there, would interfere with the wheel chocks. Amtrak could do it, but like Mike said the demand probably isn't there. They also won't take Slingshots, and even though it isn't specified, I'm sure they say no to Vanderhalls, Morgans, etc. They will take conventional trikes like H-Ds.
The auto train only operates between Lorton, VA and Sanford, Florida. Obviously they don't see enough demand for the service elsewhere to offer it cross country.