Kimber Montana in 338 Federal if I had to have a light 338. This gun takes advantage of the light for caliber copper bullets and doesn't leave as much on the table compared to the 338 Win Mag as you'd initially think. It is likely you'd shoot it better too all else being equal.
I'd go with the AWR also in this case, mostly due to the recoil. Can't say I've heard reports of a Remington Custom Shop rifle that wasn't pretty darn accurate either.
Great looking plains rifle. I worked with one a few years ago in 7 rem mag and it shot very well. Heavier than the AWR but that's for more offhand and off the sticks type shooting.
No longer made, but I'm assuming the OP might be eye balling the one in the classified's that was around $1100-1200 I believe.
What about the AWR II? I found it on the Rem site. Still made? Price?
I don't think they're worth the asking price and they no longer come with a McMillan stock. Not sure what the street price is on an AWR II, but I seem to recall their retail being north of $2500.