We have a small used car lot in Michigan. When we sell a CAR to someone in a reciprocal state, we collect our Michigan 6% sales tax or THEIR rate if it's lower; then Michigan credits that money to the other state. If the purchaser lives in a non-reciprocal state - we collect no taxes and they have to handle the Sec'y of State transactions and pay taxes in their home state.
We're on vacation at the moment, so have no idea if MO is a reciprocal state; and am not sure if a trailer follows that same (motor vehicle) procedure. I'm guessing the Michigan RV dealer is correct; although collecting 1/2 the tax will surely be difficult to report at the end of the month (knowing how the sales tax forms are filed: as a retailer, you either paid the full 6%; or you sold out-of-state and collected zero tax) And yes, to add to the confusion, some of the sales tax laws changed this month.
I want to add one more thought to the "is it really cheaper in Michigan when all is said and done?": Not sure when you intended to pick up the RV, but unless you wait until mid-April (at best) - you will be dousing it in a bunch of road salt. Granted, all of us in the north parts drag our RVs downed salty roads and they seem to hold together just fine; but it IS something to consider.