Re: 6x45 Bolt Rifle
I load 77gr BTHPs in .223 for AR-15
Can single-load 80, 85, or 90gr bullets. Not specifically "hunting", and maybe not approved in your state; but have recovered many shot at 2700fps into rocky soil backdrops and never seen one that broke apart. They bend like pretzels, but are damn tough... .02" in bullet diameter means zip in impact energy. Want to shoot deer w/a "hunting bullet"? There is the Nosler 60gr Partition or maybe a heavier Barnes or other premium bullet.
I endorse the 6mm/.22-250 AKA the 6mm International over any .308 size case as the case is only minimally overbore. There is also the 6mmXC.
A bolt rifle in .223 will do everything a 6x45 can do, if you get an 8 twist barrel. Could also go .22-250 with a fast twist barrel. For light loads, look into Cast Bullets. I have a 72gr .224 mold from NEI that is exceptional.
The 6.5 Grendel/LBC or 6mm AR also have lots going for them in the AR-15 platform. A 7mm Grendel could really be the ticket to all-around varmint & game rifle in AR-15 platform, giving you great accuracy and 8 or 15 rd magazine capacity. Easier to make an AR shoot precisely than it is to gunsmith a bolt rifle anyway.
Might look at a single-shot Donaldson Wasp or .225 win necked up to 6mm.
Best of luck and lots of fun down there in New Mexico!