Furnace Displays Heat Pump mode when AC is running

Actual, better answer:

The heat pump was definitely overheating. Since the heat wave has passed, there have been 0 issues. The ambient outside temperature is 90-95 °F this week, whereas it was 103 - 105 °F last week. This seems to make all the difference as things are purring along and the home is staying cool. Thanks for the help everyone!

The answer will depend on how much temperature drop/rise there is inside the ducts. A two stage cooling system means that the heatpump will either drop the air return temp a little bit (stage 1 is running, half cooling) or it will drop the air temp a lot (stage two is running, full cooling). Either way the fan moves the same amount of air. In stage 1 mode the net effect will be for it to spend more time with the fan on, since to hold say 70F it will have to spend a while pushing 65F air through, vs a single stage system that puts out 60F air which will get back to setpoint faster. More air circulation (assuming its all a nice 65F) should actually be better for comfort in your restricted upstairs area vs spending a short time blowing 60F air until the downstairs is comfortable, leaving the upstairs without much time to get cooled air.

If the ducts are soaking a lot of the cooling up (i.e. the downstairs is getting 65F air but the upstairs is getting 68F air) then no, the first stage of a two stage unit will not give a lot of comfort and a system pushing colder overall air (i.e. single stage) all the time would be preferable.

In any case (if you get this system or not) most ducts, even in older homes, have some sort of internal damper to control flow to each run/room. These might not be accessible if you finished your basement and covered them all up, but that's for a different question about exposing louvers and patching sheetrock ;-). Further, your vents in each room should have a shutoff (not preferable vs in-duct damper, because it's noisy) which would allow you reduce the airflow to the downstairs rooms. This would be the solution if you find that running the system just doesn't get the upstairs rooms comfortable. Start by reducing flow to the one or two ducts closes to the thermostat, and continue until your system runs long enough to cool the upstairs without leaving any of the first floor rooms too frigid.

edit: grammar and clarification