Pretty simple question to answer...
1) Take $100-200 to a gun shop, buy 2-3 boxes of deer hunting ammo for each rifle.
2) Go home and mount and bore-site your "good scope" on the .270win
3) Go to the range, zero the scope at 100yrds, and shoot several test groups with each box of different ammo, saving the targets, and/or measuring the group size
4) Return home, remove the scope from the .270, and mount it on the .300
5) Return to the range, zero the scope at 100yrds, and shoot several groups with each box of different ammo, saving the targets and/or measuring the group size.
6) Return home, compare the two sets of targets to see which load shot best out of each rifle, and which rifle delivered the most accurate (actually "precise", but that's a different matter) groups.
7) If the test firing determines that .300WM is NOT the more accurate of the two, remove the scope and place it on the more accurate .270win. If the .300WM IS the more accurate, leave it alone.
Frankly, there's no way to tell which rifle you shoot better, let alone which rifle actually shoots better. I personally wouldn't have any reason to believe that a Win 70 would out shoot a Ruger 77 (they've both had their good and bad years/eras, and neither are particularly inaccurate, nor exceptionally accurate). I'd also say that I'd have no reason to believe that either a .270win or a .300WM would be inherently more accurate than the other (although history might "rumor" that the .300WM is more inherently accurate). If I recall correctly, the "Ranger" was more of a lightweight model, so it may possibly be perceived as less accurate than a standard Ruger might be, but frankly, for hunting purposes, such as your mulies at 400-600yrds max, I don't necessarily believe either would have an advantage (other than the harder hitting power of the .300WM).
Ultimately, the human component is critical, no matter how accurate either rifle might be. Without trying to be insulting, 400-600yrds shouldn't even be on the table considering the question you're asking, so without knowing anything about how well you shoot either rifle at any range, flip a coin and you'll be just as well served as asking questions on here.