uncledjm is on the money. Your house's two-story design is the main problem as far as the overheating goes. As you know, "heat rises." Well, it's rising - right into the bedroom levels.
Adding the mini-split system will work, but in my experience those things are neither attractive nor quiet, regardless of what the manufacturers say. The do, however, tend to be ridiculously energy efficient compared to other systems.
If you are committed to replacing your system, this is what I would do if it were me:
1) Definitely go for the two-stage system. It saves money in the long run. They spend most of their time running at the lower speed which is also quieter, and it's surprising just how many things make noise in a house. Wouldn't it be nice to turn some of that down?
2) Gain access to the ductwork and make whatever changes are necessary to add dampers that will permit the upstairs to be heated and cooled separately from the downstairs.
3) When installing the new system, also switch in a new thermostat that will:
a) Permit setbacks so the system is off when no one is home
b) Add a 'stat upstairs to signal the zoning panel
c) Add a zoning control panel to operate the zone dampers
All these functions require a level of hardware not often seen in a conventional home. You're definitely going to need a tech unless you are a seriously advanced amateur.
Zoning control allows you to treat a larger home as if it were a couple of smaller homes if you aren't using the entire space all the time. If you are using the entire space all the time, you may be better off adding a separate system to treat the upstairs.