A thin and portable laptop is more expensive than a chunky one. And the thinner it is, the more inconsistent the fps. That is because they lack proper cooling/ventilation so both the cpu and gpu are more prone to overheating and throttling down. You can't get anything lighter than around 5lbs and expect it to play games well.
Just look for one with an SSD instead of a hard disk.
$1000 is a reasonable budget.
You want something with a GTX 1050. The 960m is older, slower, and sucks up more power (worse battery).
Some choices (just a quick search - I can't personally vouch for any of these but their specs fit):