For your budget and needing stainless/synthetic I would keep my eyes open for a Winchester Classic or a Remington Stainless BDL with Plastic and spend the rest of the money on a scope.
I have a Win Classic .338 and it has been a terrific shooter since I bought it new about 15 years ago. Put a McMIllan swirly on it and it is even a bit better.
They seem to come up on the classifieds here on a fairly regular basis.
Bob
Most sensible answer yet.
