"I’m looking for Hyatt options at under $100/night for long-term stay(s) this fall/winter. Can be anywhere, but must be warm-weather and open to U.S. citizens."

Ok, since you're still looking. Not sure if anyone made a suggestion of Hyatt Regency Dubia (Diera). But, the prices are cheap and the Expo is going on so you'll have something extra to do. I'm actually going to be there for several weeks. If it works out I'll stay additional weeks. It was 300 aed when I booked. But, I got globalist from staying in Cabo for 30 nights, So, I should get an upgrade and club access. Not bad for all that's included. And it's warm place for the winter. I'll be there from Nov-Feb.