I assume you plan on getting to MT via US 212. IF so, Seems US 212 doesn't come close to the Tower. But from Sturgis, it might be closer miles (but 12 minutes longer) to take SD 34 to Belle Fourche & on to WY 24.
We have driven the Belle Fourche to the Tower Route. Not too hard of one, but is somewhat narrow & twisty at times.
We have driven down from the Tower to I-90 twice. Somewhat wider roadway, not as twisting. But some.
I-90 from Sturgis to Sundance & US 14 would be the easiest route. Took a while for me to figure out the 14 you were using is I-90 & turns off to go to the tower. .
Yes, I believe it would be worth it (to me anyway) to go the route you first mentioned. Worth a stop & an overnight if you have the time.
Never been up to the US 212.