beyond 300 yards is a VERY long shot on a moose, i guide brown bears and would not let you shoot at one at 300 yards if you were my client.
i suspect that 300 is a very long shot in africa for PG as well.
as stated earlier, use a 270/or 250 gr bullet (a good quality one) and it will shoot faster out of your H&H.
6x safaris to date.. I'd guess between my wife and I we've taken 50+ PG on those hunts... Also hunted AK, Ireland, and several of the lower 48 for various "deer" over a period of close to 50 years...
and I've taken exactly 1 shot over 300 yards on a game animal in my life... a black wildebeest... which as a species is known to be very difficult to get in close on, and it would seem many/most people shoot them at distances of 200-400 yards..
I know there are people out there that fancy themselves "long range hunters" that really like doing the 300-1000 yard shots on elk, deer, etc.. but thats not me.. and most of those guys arent shooting traditional "hunting" cartridges.. they are shooting highly modified precision rifles chambered for cartridges that were specifically developed for long range shooting (like 300 PRC, 338 Lapua, etc..etc..)..
If hunting at distances of 300 or more yards is going to be a regular thing.. I'd recommend figuring out what platform you want to use first.. and then select the cartridge for the intended game (bears, etc per the OP).. I think youre going to find that your typical "hunting" rifle in any larger caliber (like 375 HH, 375 Ruger, etc) CAN make a 300+ yard shot.. but its not going to be optimum... Its not what they were designed for.. and to another posters earlier point.. they are going to seriously ouch when shot from the prone...
One size isnt going to fit all here... What I'd shoot a brown bear with at 300+ yards (I actually wouldnt.. but what I'd recommend someone else use) would be very different than what I'd shoot an elk with at 300+ yards... which would be very different than what I would shoot a pronghorn antelope with at 300+ yards... (very different terrain.. very different shooting conditions.. and very different animals in terms of size, ability to take a hit and keep on keeping on, etc..)..
FWIW Im headed out for an elk hunt tomorrow afternoon (spending a little more than a week in northern Idaho chasing them).. Ive been advised by the guy thats helping us (he lives there) that I need to be prepared for a 300 yard shot... but thats the max he would expect any opportunity to present itself... he thinks it is most likely we will find ourselves taking a 150-200 yard shot (the distance he has taken most of his elk at)..