A few things to consider.
1. Most people who shoot either 600 or 1,000 yard benchrest have a dedicated rifle for that. A benchrest stock is almost invaluable. But that 3-inch, flat forearm is a terrible thing for any other purpose. If you envision this as a carry rifle, you're starting out with a sever compromise.
2. There is nothing wrong with a Winchester Model 70. Phillip Yott, who holds the most lifetime points ever in IBS 1,000 yard benchrest, used a model 70. But it was barrel blocked, both his light and heavy guns. That makes the stock even less suitable for a carry rifle. With your action, you'll need a stock, barrel block, and a trigger. The Jewell triggers for a Model 70 cost a bit more than those that take a 700 type trigger.
3. I would NEVER chamber up a model 70, with its 1-inch tenon for a .338 Lapua. I wouldn't even chamber up a Remington 700 for the Rigsby-diameter cases. Yes, I know people have done it. The 700 will let you use a case based on the 404 Jeffrey, like the untramags. The model 70 will only if you use a slightly larger chamber than is commonly found with match rifles (Jim Borden, in an old Precision Shooting article). Accuracy should be OK though, even with the slightly larger chamber. The model 70 should be fine with a .338/375 Ruger wildcat.
There is no need to push the big .338 bullets over 2,700 2,800 fps. The .338/375 Ruger will do that, and with H-4831, a wonderful powder. Some of the biggest problems with the .338s are are, IMHO, due to powders.
4. The .300 Win Mag is a fine chambering, used by 3-times IBS National Champion Danny Brooks. Just don't get all sweaty about having to have the bullet entirely in the neck. Danny uses the standard freebore length, and 210 Bergers. Yes, the bullet is back in the case. The trophies on his wall look just as good, though.
5. The cheapest way just now is to buy a used rifle. That way, you get a benchrest stock and trigger, and if the rifle has a history of doing well, you know the action is good. Not true of all new non-custom actions. A trued 700 is fine. A Savage is OK. Plan on having to replace the barrel. Remember that dies, etc., add to the tooling costs. Maybe you can get a rifle with those included. If you want a custom, any of the usual suspects (BAT, Viper, Kelbly & probably a few I'm forgetting at 11:30 pm) are just fine., but as you say, you're in for at least $2,500 plus scope & maybe dies.
Hope that helps.
It's been said many times, but is still true: GO TO A MATCH before laying down your money.
