You might have an easier time if you look at chambering in the 375 Ruger given the shorter case length. The claim is that the Ruger will exceed the performance of the H&H but I thnk the advantage of the Ruger is in the case length and a rimless case. The rimless case is probably a personal preference. The shorter case length may make it easier to find a candidate action.
I built a 375 Ruger about the time they first came out and I have used it for black bear hunting and the two bears I have taken were impressed with the round.
Another completely different route than going with a custom action would be to go with a Zastava commercial 98 chambered in 375 H&H from the factory, rifle cost under $1,000 off of gunbroker. The Mauser 98 has one of the best track records in Africa on dangerous animals, they just work. To squeeze as much accuracy out of the Zastava you could rebarrel it with a custom barrel, that would probably provide you with adequate accuracy. For hunting appications, unless the ranges exceed 500 yards, the accuracy difference between a full custom rifle built off a modern action and a 98 built by a gunsmith who does a lot of 98's (my recommendation would be Jim Kobe of Bloomington MN) would be moot. Half MOA is not needed when shooting off of sticks as the sticks may be the limiting factor.
Understand, I am a little biased towards 98's for hunting, I do almost all of my hunting with 98's that I have built off either Zastavas or FN commercial actions, about the only modification I do to the action is install a 3 position swing safety (think Win Mod 70 safety) I prefer to have the safety lock the firing pin instead of the firing pin. I usually use cut rifled barrels, although I am building a 9.3 x 62 on an FN action and went with a McGowan barrel (button rifled) as it would have taken longer than I wanted to wait to get a cut-rifled in 9.3 mm. The Zastavas have been imported and sold under Interarms MarkX, Remington 798, Charles Daly 98's, and now under their own name all of them are fine for hunting.
I realize you are probably fixed on a modern custom action and there is nothing wrong with that, I just thought I would bring to mind a few different twists to a potential solution. Many ways to skin the cat.
wade
