The heavy barrel guns actually only weighs one pound more than an equivalent sporter barrel rifle. For example the MkII F weighs 5 pounds while the MkII FV weighs 6 pounds. I know that extra pound can make a difference especially when you're trying to shoot off hand. The other stocks add weight to the rifle too. The BV weighs 6.5 pounds. Still they aren't extremely heavy.
Both barrels will be accurate. The main advantage of a bull barrel is how it handles heat. If you're shooting a lot of varmints you'll appreciate the extra metal. If you're chasing tree rats you might wish you had a lighter rifle. The bull barrel should allow you to shoot slightly better because it stabilizes those nervous twitches more because of the extra weight. But there won't be a huge difference there.
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