My choice with your budget and wanting to shoot 1000 yards:
700 SPS Tactical. 22" heavy barrel in 6.5 creed ($550)
Remington 700, Special Purpose Synthetic, Bolt Action, 6.5 CREEDMOOR, 22" Heavy 5/8x24 Threaded Barrel, Chille Green Hogue Stock, Right Hand
grabagun.com
KRG Bravo Chassis ($350)
kineticresearchgroup.com
20 MOA Base for Remington Short Action ($60)
Buy Scope Mounts, Rings and Bases from top manufacturers like Leupold, Warne, Weaver, Talley, DNZ and more. Click through to find the latest deals.
www.swfa.com
Magpul bipod ($100)
Amazon.com : Magpul Rifle Bipod, 1913 Picatinny Rail, Black : Sports & Outdoors
www.amazon.com
Burris XTR II 5-35x50 ($1,100)
Shop for rifle scopes from top manufacturers like Burris, Bushnell, Leica, Leopold, Meopta, Nikon, Zeiss and more. Click through to find the latest deals.
www.swfa.com
MDT Premier Scope Rings ($100)
mdttac.com
Armageddon Gear Squishy Bag ($35)
The Armageddon Gear Squishy Rear Bag features a round cross-section that allows universal orientation in the hand
www.altusshooting.com
This puts you at just under $2300 with no tax, shipping, or FFL fees, which is going to eat up a lot what remains if not all of it. With anything that's left, you'd need to pick up a cheap rifle case, cleaning supplies, and any tools you don't already have to properly put the thing together (get an accurate inch-pound torque wrench, I recommend Borka).
You'll noticed I skimped on the barreled-action portion of things. This is because the barrel is a consumable anyway, so after you've put a few thousand rounds through the system (barrel life in a 6.5 creed is around 2500 rds), you can send it off to a good shop to have it rebarreled and trued/blueprinted.
If you're dying for a custom rifle and can't fathom the thought of having a stock Remington action and barrel, you should look at some of the new PRS Production-class rifles which all come in at pennies under the PRS-defined $2K limit:
Patriot Valley Arms John Hancock:
patriotvalleyarms.com
Badrock Precision Southfork:
BadRock Rifles are built for tactical and target use, the South Fork uses a precision bolt action, set into an aluminum chassis. Call us at 406-897-2776.
badrockrifles.com
Seekins Precision Havak Bravo:
<p>The HAVAK Bravo is Seekins' go-to, run-it-hard, do-it-all rifle.</p> <p>HAVAK Bravo users run the gamut, from our nation's elite SWAT and law enforcement professionals to competition shooters to hunters to those simply plinking cans
www.seekinsprecision.com
Masterpiece Arms BA PMR Competition Rifle (that name just rolls right off the tongue):
Masterpiece Arms builds World Class Rifles. We start with our industry leading Chassis System, utilize the Best Actions and Triggers produced, and a 416R SS Handlapped Barrels provide some the best shooting, most accurate rifles in existence. Designed, Built and Enhanced by Long Range Shooters...
masterpiecearms.com
Bergara Premier LRP:
The Premier Series LRP rifle features a Bergara Branded aluminum chassis made in partnership XLR Industries. This Chassis is CNC machined from solid 6061 T6 aluminum and features an AR15 style grip, adjustable length of pull, adjustable comb height, and the butt plate can be canted to give a...
www.bergara.online
Just keep in mind that if you go this route you're going to have very little for left over to complete the system. You'd likely have to find a scope for $400 or less which would end up being a throw-away item in the end.