A surprising number of cartridges will perform out to this range if the rifle is optimally throated and bullets are carefully chosen. UK and British Commonwealth 'Match Rifle' shooters use the .308W with 190-220gn bullets routinely to 1,200yd and rather more exceptionally at 1,500yd at the Coonabarrabran Ranges in New South Wales, Australia. This doesn't make .308W the ideal 1,500yd cartridge though!
The Berger 180gn 7mm VLD has a G7 BC of 0.337, exceptionally high. An MV of 2,900 fps sees it retain 1,177 fps at 1,500yd in 'standard' ballistics conditions - less in cold moist air with high atmospheric pressure, more in hot, dry air up a 5,000 ft mountain. So it would normally remain supersonic (1,122 fps under standard conditions), but not by a lot, and right in the turbulent transonic speed zone (1.3M down to 0.95M) that can cause bullets to lose speed faster or behave other than ballistic tables suggest. You'd also need a lot of elevation available on the scope with 816 inches of drop compared to a 100yd zero, 52-MOA, and 156 inches of sideways movement in a 10 mph crosswind.
Ideally you're looking to stay above 1,450 fps at your ultimate range to stay out of transonic speeds and that needs over 3,300 fps MV which you won't get from either cartridge. 7mm WSM can give over 3,100 fps in F-Class rifles and that gives you around 1,300 fps at the far end of your range which will normally work OK. The price is barrel life - even at 2,900 fps it's not good, but push it to 3,100 fps or so and it's dreadful.
I don't think you'd get that much more MV from the 7mm STW even with its bigger case as the cartridge becomes inefficient with a larger powder charge than the WSM's and you get severely diminishing returns. What does increase dramatically is barrel wear and you could be looking at as little as 250, 300 rounds before the throat is toast with the STW stoked right up - it's a one-shot, one-kill beanfield deer cartridge that isn't expected to be shot a lot.
The better answer is to go larger, not to .300 that hasn't got any really good bullets at the moment (but that may change soon, as Berger Bullets' people are currently working on a next generation family of super-high BC .30s), but to 0.338". It's no coincidence that most countries' military snipers now use .338 Lapua Magnum, the USA excepted which seems to have taken a dislike to the cartridge. Berger's 300gn Hybrid bullet has a tested G7 BC of 0.455 and is driven to 2,750 fps in .338LM military sniper type rifles, more in custom built longer barrel competition rifles. The 300 Hybrid at 2,750 fps MV travels at 1,436 fps at 1,500yd under standard conditions, right on our desired terminal velocity. Wind drift reduces to 111" in a 10 mph crosswind, around the same as the old .308 Sierra 155 Palma MK at 3,000 fps in .308W. Unlike hot-loaded 7mm cartridges, barrel accuracy life is good, at least 1,500 rounds. You still need a lot of scope elevation adjustment (48-MOA), but that applies to anything at such extreme ranges, so you might need a scope rail with a 30-MOA slope made up.
There are three big disadvantages in .338LM of course - rifle size & weight (if you have to carry it any distance), recoil, and cost. The extra weight isn't that much over what an optimised 7mm would be, the recoil is, but this is completely tamed by a good muzzle brake. These devices are banned in F-Class and similar competitions, and don't make their users exactly popular in general rifle range use, so it depends on where you're planning to shoot. Cost? If you want to shoot seriously at 1,500yd and up, everything becomes really expensive. You start to see specialist optics at several thousand dollars with external adjusters in the base etc as well as very expensive ammo costs (£3 here or around $US 4.50 per handloaded .338LM round).
Laurie,
York, England