I have a 300 H&H in the Remington Classic and my dads pre 64 super grade.
Both shoot 165s, 180s, and 200s sub sub moa. The classic shoots the 165s at 3200fps, the 180s about 3050, and the 300s over 2900fps. The Super Grade does a little better with its 26" barrel.
nothing has been done to the Winchester . The Remington has been bedded, floated, and trigger worked.
My dad bought the Winchester new and I bought the classic new in 83-84. We started reloading the Winchester in the 70's when H4831 was the goto powder and it maybe the only one it has seen. It also works in the classic. Everything from 4350 to H870 has given good results in the classic. I'd also try 4955, 7977, 8133 or one of the reloader powders. I'd put H1000 and Retumbo into the mix if they were available.
inside of about 900 yards I don't think it really matters if you shoot the 165's, 180's, or 200's. Personally I'd see if the Barnes 175 LRX, or either of the Nosler 200's shoot well and be happy.
I've been very lucky, both of my long, tapered, belted 300H&H's and my 375 H&H shoot around 1/2 moa, or about as good as I can do.