You are not gonna really get a "custom" gun for $1,200. Nothing wrong with that. I'd look at good factory rifles. My choice would be 6.5 creedmoor for off the shelf match ammo at reasonable price - I pay just over $1 a round for 140 grain ELDM's.
Tikka T3x T or T3X CTR (Georgia Gun Store carries them for just over $900)
Ruger Precision Rifle - if you are going to change anything out on it, go ahead and just get the Gen1. They pretty much just changed aesthetics and added a muzzle brake for $250 more.
Bergara B14 (Georgia company - right in Lawrenceville)
If you want to feel like it is custom, Ruger American Predator in 6.5 and drop it into a stock or chassis of your choosing, but you are getting a better barrel out of the box with one of the above and already in a better quality stock or chassis system.
Lots of folks coming out with precision rifles in 6.5 for around $1k now - Howa (heard nothing but good out of these, but very new to the game), Savage, etc, etc, etc. Lots of choices these days, but the top 3 would be my first choices
I fully understand that $1200 is a very low budget. But I figure any rifle I get will out shoot my skill set and if I ever get to where I can shoot better than the rifle is capable of I can always buy a nice rifle and/or scope.