I have a DPMS LR-308L (.308 Win) with a spare LR-308 AP4 Upper (7.62 Nato). So here is my take on which chamber to get.
The .308 Win spec is derived from the 7.62 Nato spec. They are very similar and dimensionally the same. There are some slight differences between the two rounds. That being said, in spec ammo of either type, per SAAMI, is safe to use in either chamber.
The 7.62 Nato chamber is longer and has longer GO, NO GO and FIELD gauges. This was probably to aid loading and firing for full automatic weapons. An in spec 7.62 Nato chamber should fail the .308 Win NO GO, but will likely be within the FIELD gauge. The brass used in 7.62 Nato ammo is also slightly thicker.
For me, the chamber I use is based on the ammo I use. For plinking and 3 Gun, I have a supply of British Surplus 7.62 Nato that I shoot out of the AP4. It is not reloadable and goes in my broken brass bucket. For more precision applications, I shoot .308 Win ammo out of the 308L upper. I also reload and .308 Win brass that comes out of a 7.62 Nato chamber will have the shoulders blown substantially forward and need significant resizing, shortening case life. The thicker brass also has less case volume, meaning less powder. I've been saving my .308 brass since before I reloaded, so I had a supply of it that was already fired in my gun, which was a huge plus.
For your purposes, unless you reload, either chamber will probably work fine. I would lean towards .308 Win as for hunting and the like, there is more available in .308 Win than for 7.62 Nato. If you think you will reload, you should go .308 Win and you can still shoot cheap 7.62 Nato out of it. If you find a gun in 7.62 Nato that has all the options you want for the price you want, we know a 7.62 Nato gun is probably fine to shoot .308 Win unless there is substantial throat erosion (although it wouldn't hurt to check with .308 Win gauges).
This all assumes a new gun. I would headspace check a Milsurp gun. I'm also assuming brass case ammo. I can image some potential issues with shooting steel cased .308 Win in a generous 7.62 Nato chamber.
In the 1K range you are likely looking at DPMS/Remington. Keep in mind that the LR308 with the stainless steel barrel is super heavy an somewhat unwieldy off hand as the weight is in the muzzle. The LR308B, with the 18" bull barrel is a bit easier to carry and swing.
YMMV. The 7.62 Nato/.308 Win debate can result on mile long threads. Google for more info. There is a lot out there.