Use the lathe to just remove the reeding from the dime, very carefully measure the diameter with a micrometer. Use the lathe to hollow out the rev of the cent to the size of the dime. Use a shop press to force the dime into the cent. Electroplate with silver. Of course you still have the problem that your "mule" has dates on both sides about 20 years apart, and no denomination.
Gotta agree with that. I got to see one at the local shop - a beauty for sure. If I recall correctly 20-30k coin. I think the local shop enjoys really showing me stuff outside my price range. But I do so enjoy holding them in hand.