With optics is another story,, but you could go with a Barnard,Stiller action pic a stock,, barrel,, trigger bottom metal, its not hard to get a gun with a custom action for $3000,, you might have to shop around or get to know the right people,,, but if it would cost me $6000 to get my wife to go hunting with me I would pay it LOL
first off sorry to hijack this thread. Second I have owned a Barnard and priced out stillers from many dealers. And I have bought different bottom metals. I couldn't do it for less than 4000 using the components I wanted and I thought I shopped around and got some things on sale.
To the OP if you are sold on Remington over savage your two best options are
SPS Tactical / Varmint and get better optics
or
700 P / R5 and get optics to get by and upgrade later
If you don't require or wan't stainless I would get the
SPS Tactical (on sale for 699 some places but might be out of stock) 800
Weaver Bases 15
Burris Rings 45
Falcon Menace scope used 400 ish
gunsmith worked on trigger 40
That leaves you 200 for stock upgrade the HS take off stocks pop up in the EE quite often for 350-375 ATR has some usually too for 375 so you only need to save up 150 to upgrade the stock and you would have one nice rifle and a tactical look. Upgrade rings and bases as you have money. If you can find one on sale for 699 you might even be able to get the stock right off the bat with your budget.
Best of luck