I might recommend a folding bike. They are short enough that they are all basically “step through” bikes, they are compact, light (usually), and with small wheels and modern drivetrains they can climb, though they may be slow going down hill if you don’t have a big ring on the front. You can get something like a Tern D8 with disc brakes, which has an 8 speed driveline and weighs 30 pounds. It’s well under $1000, and is easily upgradeable. Upgrading to 10 or 11 speeds and a lighter crankset is a bolt-on affair, you can easily shave a few pounds off the bike.

I used a folding bike for much of my riding, I have a Birdy, which I have had a lot of fun modifying. Its fun to ride, reasonably fast, is very light, and I can put in the trunk of my car, which is easier than putting in on a roof rack or hitch rock