Have you actually compared the weights of the bull barrel to a sporting one?
Do you really want to be trudging around carrying a rifle that weighs 10lbs+ without scope fitted?
What benefit do you really think it's going to bring you if you do get a bull barrelled gun?
Bull barrels are usually found on target rifles where they lay down and shoot from one position and they often have little carts to lug their stuff to the firing point. The Varmint rifles in the US have heavy barrels or Bull barrels as they shoot rather rapidly butt hey usually take several rifles to allow one to cool while they shoot another and often shoot from the truck or very close to it weight is not a problem when you don't have to carry it far. Of course bench rest rifles have short heavy barrels but then again they are more akin to coastal artillery and belong to my mind on a concrete stanchion mount as I believe a "light gun" is about 14lbs in weight
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Parker-Hale introduced the 1200V (Varmint) with heavy bull barrel, they claimed the barrel weighed 4lbs on it's own but did not say in which calibre and seeing as how it was offered in .308 and mine is a 6mm I guess the 6mm weighs more than the .30 cal due to the smaller bore.
The rifle sans scope and mount weighs just over 10lbs. I weighed mine ready for the range and it weighs just shy of 13lbs. Add a heavy tacticool type scope:-
and it's over 13lbs without a Bi-Pod
Now do you really want to lug that about stalking?
Now this is where bull barrels belong:-
The target range
Edit:- Load testing with a .308 Brno ZKK 601 with normal sporting weight barrel and unaltered factory bedding
It did not like the Winchester factory 180 grn Silver Tip at all. The Speer 150 grn Round Nose bullet does not suit either but the rifle like the Hornady 150 grn Round nose better, not shown on this target, it likes the 165 Nosler ballistic Tip and with a bit of tweeking the 165 Speer SP shoots well. Notice all five round groups except the Silver Tips which was obviously not going to make it after just three shots.