SA vs SA.
Same barrel weight.
Stocks are likely a wash. Both carbon fiber.
Action would be the only difference. The Bergara is a 700 clone, so it would be about the same weight as a SA 700. Not sure what the Barret SA weighs, but I'd be shocked if it was more than 6oz lighter than a 700.
I've been shocked before though.
ETA - You might have me with the bottom metal. Especially if its steel..