Hello Longwalker,
My favorite "Bush Load" in the H&H (is there any other .375 even worth mentioning ? LOL) is the Hornady Round nose soft (or "solid") at 2400 fps.
Hornady seems to be slowing or stopping production of their excellent heavy-for-caliber round nose bullets - very sad.
However, all is not lost because I've also had excellent success with the 300 gr Nosler Partition at 2400 fps.
My wife has a Law Degree and advises against posting powder charges on the Internet so, I will just say that I use Winchester brass, Federal 215 primer and the charge weight of Dupont IMR 4350 powder shown in Hornady manuals to give 2400 fps with their described bullet/s.
With the Hndy RNSP and NP brands both at that sedate speed, I have taken many animals in Alaska and Africa both.
I can hit a small dinner plate at 300 meters/yards, from the sticks with it by means of my 4x scope (with a slight "hold-over").
And, zeroed at 100, it is still nearly spot on for elevation at 200 meters/yards.
Recoil is noticeably just slightly less than factory spec 300 grain ammunition (I believe factory 300 gr ammunition is usually around 2550 fps).
At 2400 fps (muzzle velocity) the round nose bullet hits hard and mushrooms half way down the shank, "just like in the movies" so to speak.
It does not always exit large plains game such as blue wildebeest, zebra etc., but it usually does and it virtually always exits smaller PG such as impala, reedbuck, bushbuck, impala, warthog, etc., typically leaving approximately only a golf ball size exit wound.
I've never lost an animal that I have shot with it and do not recall ever having to track one afterwards either, here or in Africa.
It has been consistently so accurate that I have shot the heads off many grouse with same out to 15 or 20 paces, incident to hunting moose, caribou or deer in my home state.
Regarding the .375 and 270 grain bullet loading, I only have used it very little on anything larger than a jack rabbit and at that, only in Alaska for antlered game (and Nevada but for only vermin shooting there) and at that, I only used the 270 grain at full factory velocity (wonderful load if you might have to take a long shot across the tundra etc.)
However, I believe for hunting everything in the so called "Plains Game" category of African animals (Giraffe to duiker), in all conditions (4 yards to 400 yards), it seems to me that a 270 grain spitzer or semi-spitzer (such as the best of the premium soft points: Swift A-Frame - for instance) would be as close to a do-it-all hunting rifle as one could dream up.
Last but not least, on deer and caribou I have used the 235 grain Speer JSP which is a semi-spitzer and will shoot as flat as a .30-06 / 150 grain.
I confess to only having used it at well more that the velocity you are wanting but even with that, I did not load them "hot" and yet they were not overly destructive on meat, much to my preference.
Many people overlook the 235 grain Speer, I suppose because it is light for many Africa species, and N. America moose, grizzly or bison but, for hunting in places like Namibia, South Africa's Eastern Cape (and up here in Alaska for caribou or sheep/mountain goat out in the open), it makes a very fine long-shooter out of your scoped .375 H&H.
This morning's 4 shots of espresso are wearing off so I will clam-up now.
In closing, I submit that the .375 H&H is the world's finest hunting cartridge.
All the rest of the .37 caliber cartridges in the world combined do not offer any advantage over the original H&H as far as I'm concerned, beginning with their lack of availability in far flung places such as remote bush Alaska/bush Canada/bush Africa/Bush Northern Australia.
I always like to say "If you cannot bag your beast with the H&H version, your cartridge is fine, It is only your aim that needs improvement" etc., etc.
Cheerio,
Velo Dog.