Amazingly enough, it doesn't take much pressure to change how a barrel acts. Sometimes, yes, a barrel will actually be "nudged" by the stock, or other times, it just fools with the way the barrel moves and screws up your point of impact. Some barrels need a little pressure at the tip of the forearm to make 'em shoot. Rifles are finiky things when you really get down to brass tacks.
Realistically, I don't think you need a custom job for what you will be doing. Art is the hunter around here, but I think he'd tell you the same thing. Get a factory rifle. If you want to shoot itty bitty groups then you don't need a hunting rifle, or it will be too heavy for hunting. A real custom job will barely start at $1K...not talking about throwing a new stock and trigger on a $300 rifle, but a real, honest to goodness custom job. Recutting the chamber, recutting the barrel so its square to the receiver, squaring the bolt to the receiver, the bolt face to the receiver ring and chamber, squaring and lapping the lugs, and about two dozen custom jobs. The average Joe doesn't need this to kill Bambi. Only the man who has nothing else to dump his cash on needs this for Bambi slayin'. Now, if you've got a $1K wad burning in your pocket, I'd suggest you go to a few well stocked stores and FEEL. Pay attention to the actions, how they feel to you, how the triggers feel (they are for the most part fixable, but actions are more costly). Find out what action feels best to you, for whatever reason. Find features that you like. Decide who's wood or synthetic stock you want, of if you'll need to change it because you don't like any of them. Consider the weight of each. Wood is not always heavier than synthetic. In CO, I won't carry a 10 lb rifle+scope because I'll be putting a lot of miles down, often with cold wx gear strapped on just in case. In the east, you might be able to take a heavier rifle if you want. I like semi-autos to, but I'll be darned if I'm going to hump extra metal for Bambi. Take ALL this into consideration. Don't buy anything. Come back home (with the notes you took) and search TFL for that model. Then decide.
A lot of deer and elk have fallen to $200 beat-up gun show special bolt guns.
BTW, if money is not a huge issue, my vote would be Cooper lightweight, Weatherby Mark V Lightweight, Sako, and that (Remington?) Titanium, in that order. But see, I'm lookin' at weight and quality.