The BAT SV and the Stiller Cobra both weigh 28 ounces. The Cobra is a short Viper -- and the Viper is only 30 ounces, as is Stiller's round-bottom D'Back. Jackie Schmidt checked several D'Backs and found them one of the best machined (for trueness) actions out there. The longer BATs are significantly heavier.
With the Cobra or Viper, you can get one of Gene Beggs skeleton stocks, which is very light, and would allow easy mounting of a (removable) recoil pad. For those, I like Sorbathane. The Kick-Eez junior, available from Brownells, is about right.
Alternatively, you need a stock in the lighter region with the weight of that Leupold Competition scope. And for recoil, as others have mentioned, you can use one of the strap-on recoil vests, which means its weight isn't a part of the rifle.
Good luck