Between myself, friends and my brother we have 7 of them in 223, 308, and 6.5 Creedmoor. They are the most accurate rifles in my safe. I REALLY like the Predator version a lot better than the standard rifle. They have a little heavier barrel without being excessively heavy and are proving to be even more accurate. The added weight helps get the weight back close to standard rifle weight and reduces recoil too.
There is absolutely nothing wrong with 7-08, but it isn't available in the Predator. There isn't enough difference between 7-08 and 308 to worry about so you could go 308. Ammo is a lot easier to find, cheaper, and recoil is the same. At 500 yards the 7-08 has about 2" less drop and about 30 ft lbs more energy so it is a little better. At least on paper, no animal will ever notice the difference.
The 6.5 Creedmoor is another good option. About 25% less recoil than 7-08 or 308 and it comes very close to 270 performance on game. Ammo is now readily available at good prices, even more so than 7-08.