Oh dang I didn’t even know there was a two hose type. It makes sense though since it would be sucking the room air through to exhaust outside, creating a vacuum

Home was built in 1997 so maybe things were different! I certainly wouldn’t mind another central air unit or heat pump

Just had to solve this problem myself recently. Same issue, upstairs stifling while downstairs stays nice and cool. For a whole year I just ran the system a bit colder, but that made the downstairs freezing in order to make the upstairs feel normal. Not to mention all of the energy waste running the system for so long, my home is 2300 ft.² so I went for the dual zone with a mechanized damper. You could save on the cost by installing a manual damper that you could adjust as needed so far the system marks very well and my upstairs can get nice and cold, while the downstairs is a cozy warm temperature. I paid an HVAC company $2,300 for this to be done, but the manual damper would have been just $600.

Other options I considered though were going to mini split route like the other comments suggest, or if your HOA is strict about even that, you could even use an indoor AC or two, and have the exhaust vents blowing out the window.