If I were in your position I’d get a used X220: built like a Soviet tank,
Titleist AP2 Price, 8+ hrs on a fresh battery,
Titleist AP1 Price, less than 4 lbs weight,
Titleist AP1 Sale, and enough memory expansion room so it won’t be a dog. It’d be a bit beyond your stated budget (best price I can find quickly is $750) but it would be a nice machine that would probably last you for a while. As a reference point,
Ping G30 Sale, I have Thinkpads that do light internet duty that are pushing 8+ years old now.