I would like to use the .375-.408 to build a custom rifle. I want to use it for long range target shooting, as well as big game. My concern is giving up accuracy trying to make the rifle light enough to carry. I do not want to spend the money to build 2 rifles. I am open for opinions, what do you think?

Projectiles for target work are severely limited, the cartridge rates no mention in target shooting circles and to build it light enough to hunt it'll generate a fair amount of recoil. I doubt its worth bucking the established longrange trends ... build a Sendero style rifle using top quality components in a 7mm/30cal or 338cal magnum and you'll get what your seeking.
Cheers...
Con

Con,
I agree target bullets are limited, but seirra has a couple of great ones, so i may not need go any further. Secondly, Africa is a dream trip of mine, and I understand some areas, for some animals is .375 minimum

There was a time that the 375 H&H was used for long range target shooting.

Lobo in West Virginia