What is the closest Amtrak station to Hot Springs VA?

There isn't one that I would consider anywhere near Branson.

The nearest is Sedalia, MO, which is about 150 miles away on US 65. Sedalia is along the "River Runner" route that runs between Kansas City and St. Louis.

The other option, if your destination is south, would be to drive to Poplar Bluff, MO. That's about 200 miles away on the "Texas Eagle" route that runs from Chicago to San Antonio via Little Rock and Dallas (and then out to Los Angeles, but if you're going to LA you'd probably do better to get on the River Runner at Sedalia and transfer to the "Southwest Chief" at KC).