For 500 yards max, a 308 is a great choice. With a good bullet, it has enough power on just about anything to 600 at least. It's low cost, low recoil and ammo is available anywhere... For versatility with heavier bullets, faster is better. 12 twist or 11 twist , but no slower than that... but I'd happily run a 10 twist on any 30 caliber.
But I really don't see the need to dump a lot of money on a custom rifle
for this intended use. Some factory rifles are semi-custom out of the box and offer a lot of value these days. Just about any factory rifle bedded into a decent stock will be up to the task.
I guess adding a stock and bedding it qualifies as a "custom"; but any performance differences between the two will be extremely small at 500 yds, or even a lot farther out there.
Weatherby Vanguard, Howa, Ruger, Remington, Winchester take your pick. The Winchester M70 Extreme Weather is a little more money up front, but it already comes with a better stock than the others do.
What can't be skimped on is good quality optics and that costs some dough...no way around that. I'd reassign the money saved and invest in good optics and reloading equipment...those are absolute essentials in my mind, and still be money ahead.