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Here's a crazy idea..how about you just use 5.56 like St. Stoner intended and give up the bullshit calibers for the AR?
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Please ignore the comment above, I'm sure it was meant for GD or something and this guy just accidently clicked the wrong thread.
Which caliber really depends on your intended use. I've got SBRs in 556, 300, and am working on a 6.5 right now. They each have specific uses, pros and cons, and roles.
For cheap shooting at the range, 10.5"-11.5" 556 is the winner. You can have a build knocked out for under $1000, stamp included, and the ammo is dirt cheap.
For suppressed awesomeness and hunting at shorter ranges (max range of 250ish yards, depending on ammo), 8.5"-9" 300 is a blast. The rifle is cheap to build, likely within the same price range as 556. The ammo will cost more though. Plinking ammo is down to about $0.50 a round right now for range ammo, and you could reload for cheaper, but long days blasting away will put some hurt on the wallet, with 556 being nearly half the cost per round. (comparing wolf gold 223 @ $0.27 CPR to S&B 124 Gn 300 @ $0.51 CPR). With subsonic loads, these rifles are crazy quiet and a real blast to shoot, and some would argue capable of hunting even subsonic to traditional pistol ranges.
The last option is a 12ish" 6.5 Grendel. This would likely be the more expensive route to build up front, but, wolf steel 6.5 ammo is about $0.27 CPR for range ammo, and should have hunting power out to around 400 yards with good hunting loads. It is going to suppress about as good as 556, and nobody is doing subsonic loads that I know of with a whole bunch of success. So if you want crazy quiet subsonic suppressed fun, look at 300 BLK.
Personally, i'd likely avoid the 338 Spectre for a while. Ammo costs will be higher, parts costs will be higher, and, from what I've seen, there just isn't a ton of support out there for it. Its more of a wildcat cartridge, which is super cool, but not my preference. If this concept interests you, i'd likely suggest going up to its bigger brother, the 458 SOCOM.
The above are solely my opinions, and i'd suggest you do a LOT of leg work with your end goals in mind before committing. Or just build them all...