You physically will be half way to SD going to Michigan for the purchase. I would consider going on to SD with both rigs, and shifting property from one to the other there as well, as establishing domicile and getting other chores done at the same time.
Your issue likely is getting a temporary tag to arrange title in either NY or SD.
I dont know if Mi will give you a temporary tag , all I could find online was for residents only.
If your goal is SD for all paperwork, have you already established an account with Dakota Post or another service. For a fee, they will get your title and plate done and then mail everything to you. You would be stuck in Michigan for a few days waiting for the paperwork and tags.
Good luck