Call both the Missouri and Michigan Secretary of State's and get the real info on what are the laws. Michgan sales tax laws have change slightly sense 2010 and for the better to the consumer/and residents. Sales tax in Michigan is 6% on non-food or prescription items. 50% is something I hadn't heard about but I sure would certainly check it out. If 50% tax is on each end it would seem right.
Years ago we bought a 16' boat, motor, and trailer in Michigan and paid the Michigan sales tax and the license bureau gave us and "in transit" tag for a license plate. It was going to stay in Florida and when we got to Florida they tried to charge sales tax again so I called my sis-in-law who worked as a manager at the WPB Florida IRS office and she had to read the printed law to the clerk at the Fla driver's license branch. Michigan had a reciprocal agreement with Fla that taxes only would be paid at time of sale in either state. The clerk said she didn't know anything about that law, DUH! We paid nothing in Florida but the state trailer tag and the boat stick-on tag and couple dollars for the filing fee. I don't know if Missouri has this same type of reciprocal agreement with Michigan.