But if it was you buying this rifle, a custom rifle, considering money is not a real option up to 5 or 6k at least, what would it be.
I was in a similar situation back in 2010. I wanted a rifle for everything. Fancy enough to make me feel good, durable enough to serve me for the rest of my life and most of all, reliable and accurate. The definitive rifle for myself. I ended up choosing Weatherby MarkV action for the project, mainly because it can handle even the hottest handloads with ease, 9-lug 60° design is quick to operate and I managed to source one in .375H&H with unusual (for Weatherby) matte finish.
What really sets it apart from everything else is the stock. I contacted Accurate Innovations for a 100% custom job; I selected a very nice exhibition grade english walnut blank and had AI build a slightly modified MarkV-style high comb design with customized pull length and drop. Externally it's very traditional high grade stock, but it has an aluminum frame inside the walnut for extremely precise bedding. I didn't want a muzzle break so I opted for a mercury recoil reducer and 1" Decelerator pad. I was prepared to have the action rebarreled, but there was no need. Once assembled, the rifle printed consistent .7-.9" groups at 150yd with Weatherby recommended factory soft points and just over an inch with Norma solids. All it really needed was very slight smoothing out of the bolt and trigger surfaces. I've been experimenting with different scopes and it currently has a Zeiss HD5 2-10x42 as a compromise for big game and medium to long(ish) distance hunting. Rapid-Z reticle isn't ideal for all purposes but I'm getting used to it and it makes longer shots a bit easier.
The whole project set me back about $6k when everything was said and done, which was a fair bit less than I originally expected.
All in all, Accurate Innovations' customer service is first class, fit and finish is second to none and aluminum frame and bedding makes the best of a quality action regardless of ambient temperature and humidity. Originally my choices were MarkV, Sako L691 or Mauser-style action, all of which are great for building a custom gun. All of them are available in left hand configuration if need be. My son is left-handed and I've been thinking about having a similar rifle built for him as well.
As far as caliber choice is concerned, I'd probably chosen .300WinMag over .375H&H had there been no need to hunt in Africa. Bull elks can grow fairly big so you might want to consider the old "bring enough gun" -principle, even though hunting deer with a .375 is a bit overkill. If you handload, you can always adjust your loads according to game, too.