I use 270 gn TSX in mine and my boy (#1 son) uses 260 ABs in his Ruger, both have proven to be excellent bullets on game. I load RL 15 behind my 270 TSX for 2900 fps, not all rifles will get this velocity but my 700 does with a 24" barrel and the load is 100% safe and has been used in 55 deg heat in Zambia without so much as an ejector mark.
I know fellows who use the 235 gn Speer for light game and have claimed great success with it, but I don't believe it's an "eat up to the hole" bullet. I suggest you play with it a bit and try some bullets and powders THAT YOU CAN GET NOW, you'll probably find, like most that the .375 H&H likes almost any bullet and any med slow powder, right from IMR 4064 to H4831 SC with heavy bullets. Not a complicated cartridge at all, and a real friendly reloader number.
I'm working with a 285 gn Matrix in a 375 Wby lately and a 300 gn Part load for another H&H I have, but haven't been able to make it to the range in the past month.