Well, if you're looking for a #1 in .300 H&H, you're either gonna have a long search with an expensive end result or gonn ahve to wait a year or two til a small run of 250 is shipped as part of the craig Bonnington series. The ones already out there are few and far between, highly sought after and will be a weee bit pricey (think $2k+ for a minimum). That said, a .30-06 can easily have the chamber re-cut for the fantastic .300 H&H cartridge. Gonna do that to mine.
The .375 H&H model is quite available and should be easy to locate or order. Either will be fine. remember these have been widely used in Africa for all those big creatures so a North American anything is gonna be quite do-able with either as well.
Something to consider is ammo availability. The .300 H&H isn't nearly a widely found compared to the .375 H&H ammo. If you're handloading, this is a moot point.

Myself, I'd go with the .300 H&H just cause it's classic and different.