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Hi fabwing, I can think of a lot of factors I would consider when choosing a .308 win barrel profile.
Length: since 300 yds is fairly short range a 20″ barrel would give ample velocity. It is also short enough to be manageable with a sound moderator attached. A short barrel also allows for a heavier profile for the same weight.
Profile: The emphasis on barrel heating or portability will depend on how much stalking vs. target shooting you plan to do. A common theory suggests that hot barrels tend to whip more, possibly changing the point of impact. I can't say how much heat can influence a given barrel, but it is not my primary consideration. Heavier profiles are stiffer and usually exhibit smaller changes in point of impact due to mounting moderators. medium Palma profile (5.35 lbs/30″ for example) might be a decent compromise between portability and stiffness although with a short barrel you might manage a heavier M24 profile (6.2lbs/26″ for example).
The stock you use will have a large influence on the handling characteristics of your rifle (you will need to modify or replace the factory stock to fit the different profile).
I have experience with other than factory profiles (heavy and light) in the Tikka line, so don't take my observations as recommendations so much as just adding to the conversation.
My one recommendation is to find a reputable gunsmith to do the work – there is no cheap way to get quality work in this craft!
Good luck! (And look at a CTR befor you decide!)
-Eric
