I have all three, .308Win, .260Rem and .243Win and also the .223 and .30-06 bookends.
I am partial to the .243Win as it is more versatile for varmints and paper and you can get a flatter trajectory and more speed (wind calls)...BUT, at least as of today, there is only 1 true target load and it is $2/round. Then again, there are only a few rifles that have a twist capable of utilizing the 105/107/115 loads where it really shines. So, much as I like it, when paper is in the mix and reloading is not on the table, cross out the .243Win. People balk about barrel life, and if you hot rod it, yes, 1500 rounds. If you are reasonable and don't use the speed unless you need it, you can easily get 5-6K out of a .243 barrel, but then again, better suited to reloading.
The .260 is a great cartridge and it goes up and down in popularity. Ammo is readily available for both hunting and match grade at reasonable costs. In fact there are more loads available for the .260Rem than the 6.5CM. I have shot matches with targets out to 1200 yards with it and game out to 700. Of the three, this would be my recomendation based on your criteria.
The .308Win, IMHO is too close to the .30-06 bookend and the recoil is higher defeating a few of your stated desires. It is more expensive to shoot, especially with match grade ammo. I would not even consider it. My boys shoot an Encore in .308 for Elk, but the older one now uses a .30-06 and a .243Win, so will the younger in a few years.
I would not suggest the .30-06 based cartridges as the weight goes up and no need for it. I would also stay away from the 6.5x55 and the 7mm-08 for the ammo availability. If you reloaded, they could be in the mix.
Much as it pains me to suggest a cartridge I do not own, the 6.5CM is likely your best bet. When Ruger chambered it in several of their rifles they gave the 6.5CM a huge boost. You can get really good hunting and match ammo for it without reloading. It is very similar to the .260Rem and was actually designed to work better in gas guns than the .260, but it has a solid foothold with bolt guns as well. There are some new loads coming out in 2017, and I would bet a few more guns chambered in it. Based on your criteria, the 6.5CM is the best choice to fill in your gap between the .223 and .30-06.
IF you decided to reload (you should) then you could really dissect all the nuances of the 6mm and 6.5mm cartridges and pick the rifle and cartridge that you think is the best for you.