If we just use some typical values for high class factory loads, assume 1000 yds drop, and a 10 mph cross wind.
A Berger 6.5 CM 153.5 LR Hybrid leaves the muzzle at 2702 with a G7 BC of 0.356, drops -306.45, drift 61.48
A Berger 308 WIN 185 Jugg leaves the muzzle at 2608 with a G7 BC of 0.283, drops -379.57, drift 88.78
Those numbers speak for themselves in terms of which of the two will have a ballistic advantage on the range. In each of your rifles, getting high quality bullets, with a higher BC, is typically going to be the key when going out past 600 yards. In 308, 175 and up unless the rules restrict you. In 6.5, roughly 140 and up.
That said, if you shoot 308 in Palma, there are going to be many folks up and down the line also shooting 308 Palma, and they shoot to 1000 yards. Rules can vary, so check with the clubs near you.
If you shoot 6.5 CM, you will be going up against folks in open class, who will likely not be shooting 6.5 CM. The 284 WIN and the derivatives are very popular for highpower and F-Class at 1000, and then you see the short magnums and magnums show up too.
My advice is to shoot what you got, as much as you can, as far as you can, and learn wind. Getting ammo put together is difficult these days so what you have in hand is more important than what is better on paper in the short term. As you plan forward, go study what the better shooters are using in the games you are interested in joining. If you stick with those guns, then you are looking for the higher quality, higher BC bullets.