I doubt if one would ever "need" a bullet heavier than 270 grs for North American game, but I like the way heavy bullets work. The 300 gr is the traditional bullet weight for the .375 H&H, and if I could only shoot copper cup - lead core bullets, the 300 gr Hornady would be my top choice. From the data I've accumulated for the .375 H&H and the .375 Ruger, 350 gr bullets are about as heavy as make sense. The 380 gr Rhino was developed with the .375 H&H in mind, but I don't think you can get quite enough velocity from the H&H for this tough bullet to perform optimally. Loaded in a big case .375's like the Ultra or the .378, they are excellent. The 260 gr AB, loaded to 2700+ fps is an accurate, hard hitting, flat shooting, bullet and would make a good choice for the one rifle hunter going after medium to large sized game where a shot might be on the long side.
If you ever have the need for a solid, those that are short for caliber with parallel sides and a flat nose seem to perform the best. The Barnes 270 gr Banded Solid would be a top choice. The traditional Kynoch style solids for the .375, typified by the 300 gr Woodleigh's and Hornady's with their tapered sides and long length tend to be unstable in penetration. I still have some 286 gr PMP solids, another mono-metal solid, but with a hemispherical nose rather than a flat nose that also performed very well in tests. While in Tanzania I shot, and saw game shot with 300 gr Trophy Bonded solids which are also flat nosed, parallel sided bullets, and despite the fact that they are on the long side for a solid, they performed perfectly on a variety of game.
The .375 H&H does well with powders with burning rates consistent with 4350 (H-4350, IMR-4350, Win-760, Re-15). Hodgdon is marketing a new powder called Hybrid 100V that has a burning rate similar to RE-19. When I was working up loads for the short barreled .375 Ruger Alaskan, this powder gave the 260 gr AB better velocity than any other powder I tried. I haven't loaded H-100V in the H&H, but I don't see why it wouldn't work as well as it did in the Ruger. H-100V is now my go to powder for my own high performance loads in medium and large capacity cartridges. Google St. Mark's Powder for more information about the Hybrid powders.