To me the obvious thing to do would be to rebarrel the 300 Win with a custom barrel.
You already said you like it, and if you load for it you are set. The action if not cheeper in quality
so the reason it was cheeper was the stock, the finish, the trigger ETC.
A custom barrel and installation + bedding should cost less than The price of a new rifle in the
$700.00 range and definitely out shoot the existing rifle as it is.
I prefer the 300 RUM ,but if you already have a 300 Win Mag there is not a lot of difference
and the Winchester is certainly capable of long shots if fitted with a good barrel, stock and a
good trigger.
If you are set on a 6.5, Two come to mind. The 6.5x47 has a rep for being accurate. And the 260
remington (A 308 necked down to 6.5)is powerful, fits in short actions and ammo can be bought
almost anywhere at a reasonable price.
Just my opinion for what it,s worth.
J E CUSTOM