I'm from Michigan originally, have bought there for licensing in another state, and have bought in other states for licensing in Michigan. The rules are different for each situation. You need to put the two states together to get the right information.
Most recently, when I moved back to the U.S. from China, I bought a new car in Michigan (Detroit area, clearance time, and my port of entry). Because the car was going to Oklahoma, there was no sales tax to pro-rate (Oklahoma collects no sales tax on motor vehicles). Michigan thus collected no sales tax; I left with bill of sale, manufacturer's certificate of origin, a "temporary" Michigan and a 30-day transit tag. The dealer had to get the tag and temporary title from Michigan Secretary of State office before the vehicle could move off the lot; that's Michigan law for new car sales. I had never heard of a "temporary" title before, but had previously been required to obtain title in state of purchase before moving it, 35 years before.
The first car I bought in Michigan, paid sales tax, titled and tagged it. I ended up paying Oklahoma penalties because I drove it too long on Michigan tags before re-titling in Oklahoma, but back then Oklahoma was collecting property tax on motor vehicles. Rules have changed.
But your situation is different. Missouri has sales tax, probably has a sales tax compact with Michigan. Michigan may not treat a TT as a motor vehicle (Oklahoma certainly doesn't).
Start by asking in Missouri (Secretary of State, DMV, State Treasurer, whoever handles it) what you need to do about taxes and title. Take that information to the dealer in Michigan, because he likely knows only Michigan rules. You do the wrong thing in Michigan, it could cost you more in Missouri.