Ive had/got both heavy and light barrels.. Its horses for courses.. When load developing on a range, a light barrel is a pain although it must be noted that a heavy barrel takes longer to cool..
Out stalking, well with a heavy barrel, you warm up faster after lugging it around! I personally dont feel a heavy barrel is needed for stalking. Even in cull situations, shooting 3-4 deer in the space of 2 minutes and maybe 20 shots in the space of an hour, i find that a good quality barrel doesnt wander all over.
On my predator i went for a medium barrel, half way between a remington varmint profile and a standard tikka sporter. Its not too bad to carry round and allows 5 shot groups to be fired without the worry that the group will open up at some point between shot 1 and shot 5.
In fairness, the difference between shooting cold bore to shooting it hot with 5 round groups is less than 0.5MOA so not a worry for stalking.
Ive walked for miles up hill and down dale with a varmint profile barrel and i wouldnt do it again.. unless your shooting dozens of shots one after the other, go for something which is pleasant to carry!