To chime in on this, I own a Win Model 70 in 375Weatherby, it is a classic stainless model that I had rechambered several years ago.
I actually fireform my brass from 375H+H brass with a fast pistol powder and a plug of soap in the neck, works wonders and produces brass that EXACTLY fits my chamber and has no stretching above the belt, which did happen to me when firing full power H+H handloads to fireform with. The .001" difference poses no problem, nor does the difference in apparent headspace dimensions between the 2 cartridge specs, once fireformed the shoulder takes over for this duty anyway negating the belts official purpose.
My brass lasts up to 10 loads, has very little stretching and runs a genuine 200+fps faster than the H+H runs with all bullet weights. I was very impressed with the reduction in case stretching, it has almost gone away. After I fireform my brass it generally shortens around .010"-.015" in the neck and this has taken 4-5 loads to stretch to minimum length of 2.840"
Yes, a RUM reamer will totally clean up a H+H chamber if the neck/body juncture isn't oversize in the original H+H chamber, some rifles have been known to show a step just forward of the shoulder.
In all honesty, the difference in performance between the RUM and Weatherby is neither here nor there, we ran 2 rifles side by side and with identical bullets the RUM only averaged 20fps faster than the Weatherby, even though the RUM had a 26" barrel compared to my 24" barrel.
Hope this helps you choose one.
You will need to clarify whether you're referring to the 375 or 378 Weatherby to get a correct response from us mere mortals.