I'm going to ---- alot of people on here off, but to do what you (op) wants to do is a complete waste of money to "build" anything. There are any number of rifles under $1000 that will do exactly what you want and they exist in the cartridges you list. And if you happen to get the occasional dud-in-a-box, sell it and buy another one. You're still ahead money.
Before any of you blow your tops, or computers up, my friend (and gunsmith) and his neighbor were sitting around talking 2 days ago about this. He does this for a living, and he's as ultra precise as a brain operation with a laser. He looks me in the eye and says, "You know as well as I do many of the off-the-shelf guns will shoot just as accurately as most any custom gun." I admitted MORE accurately many times. He grinned and said he hoped none of his work fell in that category. I grinned back and said not so far, but the jury was still out on the last one. A $250 Marlin XL7 or XS7 will hold .700" @100yds all day. That's why Remington bought them and ceased production. They had a decent trigger and cheap but utilitarian stock and a fluted bolt, along with an excellent safety. A Bergara low end model will shoot in the .5s. How do I know? Most of your Browning X-Bolts will shoot half or sub-half MOA without buying the McMillan version. I have multiple examples of that. I'm making my cousin some ammo for his now-discontinued Remington 710 in .30-06. It was considered by most to be junk. The barrel isn't free floated, the factory trigger is 6 lbs, and it ain't much to look at. That stupid thing that sold for $300 new shoots one ragged hole with my handloads and a good rest. I've got guns I've invested $4000 in that won't do that, man! More than one! I could rework his trigger for him and buy that gun for whatever he wanted and be REAL happy! Winchester XPRs will go at least to the .7s, also. You want something special, invest $700 in a Sauer 100 in your 6.5 PRC. Or if you just like the cool, tactical look, buy the new Ruger Hawkeye Varmint Target rifle for $999 in 6.5 PRC with full 26" tube. It has a 2+ lb trigger in it and demonstrates sub-1/2 MOA in reviews. The first guy that commented on this post advised you to buy a Tikka Tx3 and have it rebarreled. I'm telling you that you don't even have to have it rebarreled. The Tx3 will shoot 1/2 MOA with it's preferred load off the shelf.
There are only 2 reasons to build a custom gun. Well, 3. Ok, 4. 1)For a particular style that fits the shooter better than available models, 2)For a special caliber (cartridge chambering) that isn't available in the desired model or anywhere, 3)For a competition grade rifle, and 4)Just because you want to and what people like me say is boring and can't possibly be true. Then $60,000 or more later after 20+ custom guns, many of them wildcats, have been built and paid for, you realize only one or two was as great as you thought it might or ought to be. Yes, that's me. I've easily spent money to build at least 20 custom guns. I've used barrels from McGowen, Mullerworks, Shilen, Hart, Pac-Nor, Krieger, Bartlein, Douglas, Brux, Lilja, Wilson, etc. and had as much blue printing as any machinist would care to do on Winchester, Remington, Pierce, Weatherby, Savage, and CZ, and spent several thousand dollars in parts, labor, and loading components for each one. Not one was capable of doing what the $300 Remington 710 is doing now. One other gun was capable, and it was a Browning A-Blolt Eclipse in 270 WSM. Now I've had some quarter-inch big magnums that were capable of a whole lot more than this .30-06, and I've had some very capable .22 wildcats.
Look, if you want to build super, guaranteed accuracy into a custom rifle, you can. It's going to have a near straight-taper barrel sitting in a Manners, McMillan, or AG Composites stock. If you want to tote a 15-lb rifle into the woods every time, it will be superbly accurate. I believe what you are asking for is something you actually enjoy carrying and not "lugging" or "hauling" around. Go to
www.stockysstocks.com and you'll be able to configure the stock you want. If it's not available, you'll know where to look to get it factory direct.
www.bugholes.com and
www.redhawkrifles.com ALL have precision parts available. you're dead set on forking out the dough you could be spending on high-end glass, my advice is to see which stock on Stocky's site tickles your fancy, and order it. Go over to Red Hawk Rifles and buy a Douglas Air-Gauged barrel in appropriate contour for your wants and specifications in your bore size of choice. Have a knowledgable, capable, and ENTHUSIASTIC gunsmith do the machining and bed your action choice in your stock of choice with that Douglas air-gauged barrel and be prepared to spend a lot of time at the reloading bench to figure out the gun's desires. That's right. You HAVE to reload to get the performance you're looling for. And top accuracy. Don't blame the rifle for poor accuracy if you haven't taken the time to craft and feed it what it wants.
$1800 is going to be about the least you'll spend for a custom buy a Browning X-Bolt in 280 Remington, or the new Ruger Target in 6.5 PRC, or the Sauer 100 in 6.5 PRC, or a Savage Long Range Hunter in 6.5x284 and be amazed at the accuracy and value in what you get- or throw it out and buy another one if it doesn't function up to par. I could buy 6 of the Rem 710s for my least expensive custom and the extra dollars bought no more accuracy. They bought perfect balance, good trigger, extra long barrel for extra velocity, and unique caliber/cartridge- and great accuracy- just not quite as great as the 710 model. AG Composites and Manners are my 2 favorite choices for custom stocks. You won't go wrong with any barrel you pick from any one of the Web sites I listed. It's easier to just go ahead and buy an aftermarket action rather than make smith do all the dialing in to true/blue-print a factory action. Lots of big name, flashy actions out there now and list grows by the day. Still nothing wrong with a plain old Stiller Predator to drop into a Manners stock and have it properly bedded. I have a smith I'd trust to do anything here right by home and his name is Johnny Garris and he owns Custom Gun Coatings and Repair in Mobile, AL. He can do anything you want done and do it faster than 90% of any other smith in the country. Most of your "big time" smiths are only interested in building you a super-ultra-long-range-badass-countersniper-tactical-match rig weighing the requisite 15 pounds or more capable of shooting about what the $300 Remington 710 is shooting. A "regular" hunting rifle is beneath them. Tell Johnny that Ray Young from Hurley, MS sent you and you might get an even better deal than he's already going to give you.
Another option is a semi-custom. Montana Rifle Company and Nosler both make sub-MOA rifles in the calibers you specified for half the price of a Cooper. If you want to go a little higher up the food chain, buy a Fierce Fury for $1999 and it's GUARANTEED to shoot 1/2 MOA! Best deal on the market, IMHO. But don't just throw money at the idea. You are much more likely to be disappointed since your expectations will be perfection for something "custom" you paid all that money for. Good luck!