The mini-split route idea is a great solution, but I am wondering if you have an unvented or poorly insulated attic that is a contributing factor to your problem. If your attic space isn't setup properly and you throw a mini-split at the problem, it is sort of like pumping water out of a boat without fixing the leak that is flooding the boat. Even something as "simple" as a powered attic vent could help in HVAC performance and operating costs. If the hot air can't vent or your insulation is inadequate, it just stacks down and heats the top floor.
The other thing I would be looking at is if the HVAC run up to the second floor is done properly or if the dampers are not right. A house that size really shouldn't have much trouble if everything is balanced right.
Closing vents on the first floor and leaving the second floor open will increase the wear on the fan and may decrease efficiency if not enough air can be moved through the open vents. Knowing how bad an idea that is would require some measurements and math.
Having the nest with a second temp sensor would just force your HVAC to work harder to cool the upstairs. The single sensor nest can be a smart move to reduce costs with the higher degree of control.
Having dampers installed does more or less the same thing as closing vents, but with a higher cost. A zoned system is expensive and kind of silly on a house of that size.