6.5 Grendel as a 600yrd elk rifle?
Not even 260rem in my book, and not likely a .308, but I'd take it. (For the record "matching ballistics" is a very loosely used term. My I have loads out of a short barreled .223rem that are almost a 'ballistic match' to my .30-06, but that don't make it good for deer hunting. ENERGY people, not just trajectory).
The reality of it:
Carrying a 13-17lb rifle into the field sucks, even if you're hunting from a stand. (I do it quite often, and recognize why)
Hunting at 600yrds+ requires a LOT of shooting at 600yrds+ to practice. Wind conditions, temperature, elevation, humidity, etc all factor in so much at that range that ensuring clean kills requires a LOT of dedication at the range.
600yrds 1MOA = 6". At best you're looking at 'somewhere in the lungs' with a 1MOA rifle. Keep in mind that a 1/4MOA rifle at 100yrds may not keep it together at 600yrds (rather you likely will not be able to keep it together). You really need something that you can put up 1/2MOA groups at 600yrds off the bench, recognizing you won't shoot that well in the field at go-time. (1/2MOA @ 600yrds = 3" group = just a touch smaller than the size of a deer or elk heart). Realize that before you send a bullet 700yrds after game animals.
Making shots at 600yrds requires a solid rest, which means your set/stand design will be different than conventional hunting styles. Most of my longer range sets are designed to let me lay prone either off a X-bag sand bag or a shooting rest, or shoot off of a bench from the same (in a blind). 600yrds+ off of sticks is a VERY low percentage shot, 600yrds+ offhand is only common by outdoor sports writers...
You absolutely need GREAT glass at that range. Leupold VX-3 LRT's are about the cheapest in the group, Bushnell's 6500 Elite 4.5-30x50 is pretty reasonable. Nightforce can't be beat, but the scope will cost more than your budget.
Getting a 600yrd+ rifle for $1000-1500 isn't realistic unless you're talking used, or getting lucky. $2500 should get you there, but might be iffy. You should plan for about $1200 into the rifle (with prices right now) and another $1000 into the glass. Shooting rest not included. A bulls X bag will put you back about $100 these days, and will work well enough. Bipods are decent, harris or nothing in my book, but sand or solid rest = better. Rock BR = great front rest.
Factory ammo CAN shoot that well out of a factory rifle and kill game at 600yrds+. Is it likely or common? No. Since you'll be mobile, setting up a reloading system probably isn't in the stars.
I'm a huge fan of Savage 10 and 12 for factory accuracy. Rem 700 Sendero's will cost you more than your budget I believe, but that's my other factory favorite for long range precision. At $2500, I don't believe you could get a proper custom job done for long range precision work, not with satisfactory glass.
A rifle that surprised me a lot: Savage 16 Bear Hunter .300WSM. All the savage accuracy out of lighter barrel, same high performance Accu-trigger, and a very reasonable pricetag.