When it comes to long range shooting I've never seen a guy at the range with a rifle he didn't at least change a little something on here or there. Your first rifle for long range is going to be your learning gun. Learning not just about long range shooting, but also about learning what you like and don't like about the rifles fit & finish.
You menioned a lot of Remingtons and I'd suggest that you also consider Savages. If you look at the 12 series and law enforcement series you will find plenty of options in a variety of calibers. I'd suggest going with the 308 just because there is so much support for it out there.
Once you get a feel for your rifle and you've been shooting for a while, then it may be time to consider a custom gun. It's kind of like turning 16 and your first car is a dodge viper without trying anything else first. When I knew I wanted to get serious about long range I called savage and made a special order for a 110fp in a different caliber, switched out the stock and I'm the only guy with my rifle. My savages factory accuracy is just as good as some of the custom guns I've seem on the range if not better. You shouldn't have to pay thousands to get a rifle shooting less than 1/2" @ 100. It's not necessary, there's nothing wrong with it, but it's just not necessary.
Learn long range shooting first and find out what you like then get into specialized guns and calibers.