There is no one best long-range cartridge in a short action because we all have our own individual preferences, and needs based on where we hunt. In many places in the east a 200 yd shot may be long range because of vegetation or population conditions. Whereas out west it can vary from eastern-like distances when west of the Cascade Range or in northern Idaho or NW Montana due to thick veg conditions, or the wide-open spaces elsewhere in the drier areas of the west.
That all being said I just completed a custom M700 in 6.5 Creedmoor. While I really can't use it's capability being here in NE Minnesota but when I get back out west (were I lived previously) it will be my short-action rifle for open-space hunting. But I also have a M700 custom 6.5x285 and CZ550 6.5 Swede to use but those are on long actions.
My 6.5 Creedmoor was built as follows.
New M700 SA - blue printed 26" Lilja ss Senerao contour 1:8 twist fluted barrel Old-style Remington VLS stock Tuned factory X-Pro trigger.
I've got a Vortex 6.5-20x50 parked on it using TPS rings and a LaRue Tactical base.
The shop that built it is Little Crow Gunworks in Spicer, MN. He advertises in Handloader & Rifle magazines. Nice guy and a competitive shooter himself.
With the Creemoor I'm getting 0.5 MOA groups (using factory standard & Superformance ammo) with it abeit at short 100 yd range due to my range limitations in the area. Will be getting into load development soon but may be limited until next year because of the nasty winter weather here.
Hopefully in a couple years I'll be back out west to truely tune the Creedmoor for longer range hunting.
BTW - I'm really not into max velocity rounds. I don't hot rod my 6.5x284. I look for inherently accurate cartridges, develop the most accurate load for it, and kow where it shoots. Then while hunting I try to get into the best position possible for the shot. To me that's why it's called hunting.
I've lost count of the number of shooters I've seen that can't shoot their uber-mag/hot-rodded rifles, and once off the bench they're even worse. Plus I've seen my fill of hunters out west that take longer shots that they are capable of because they think the rifle makes up for the lack of getting a close as possible for a reasonable & humane shot. Sorry about the soap box.
This is one of those impossible to answer questions. There are many factors that could enter into one's decision as to the best caliber for short action and long range. If all you care about is the ability to hit a paper target at the maximum possible range with the least bullet drop then the decision is very easy, just look at the ballistics of the cartridges with the bullets you would use. But if you want to consider cost of ammo, availability of ammo, availability of ammo components (bullets, cases, etc.), the availability of reloading data, the energy in the bullet at a particular range, etc., the decision become much more complex (as well as subjective). For example, 22-250 or 220 swift might be the best possible choice for flat trajectory out to several hundred yards, but it surely isn't as powerful as .308 at 500 yards (or any other range for that matter). Many people like the .223 for it's low recoil, and wide availability, and some people even use it successfully at 1000 yards with very heavy bullets, but I doubt anyone would claim that it is the best. .308 is probably the best possible choice from the point of view of cost and availability, but some people find the recoil to be too high (while others use a 50BMG for long range), and the trajectory way out at 1000 yards and beyond is certainly not the flattest of all short action cartridges.
Decide exactly what you want to do with the rifle and what your budget is (in time and dollars) and then you can make a rational decision on cartridge. I once saw an article where the author looked at the overall cost of owning a long range rifle. He found that if you included the cost of ammo, required to stay good enough to actually be able to shoot long range, that the ammo cost was several times more than the cost of the rifle and barrels (I think he figured the barrel life at 4,000 rounds).
Not much to learn with AI. For example, with a 243 AI just shoot regular 243's and presto you have 243 AI Brass. Then load them with your AI die set. If you forget your bullets on a hunt you could still shoot regular 243 rounds (checking zero first). A 243AI or 260AI would be a relatively easy and affordable conversion for you.