In the price range your talking about for the 338 class rifles there is pretty much one option , 5.5-22 Nightforce . Unless you find a smoking deal on a used S&B thats what I'd get . Get the hi speed knobs and zero stop and either the MLR2 or MOAR reticule depending on your tastes in graduations , Mil or MOA .
This is what I'm running on my precision rifle. I've shot it a few times at a grand and it worked very well. If I remember right, I was able to almost recognize my shooting buddy's face while he was resetting a target. ( the bolt was in HIS pocket)
"It takes considerable knowledge just to realize the extent of your ignorance"