If I were having a British gun built I would go with .300 H&H Flanged. If European, then 7x65R. Either of those would be very versatile. However, you might consider a two barrel set. A 300/375 H&H Flanged set or a 7x65R/9.3x74R set would cover everything in America. The extra barrels will add about 15%+/- to the cost but you'll be set for anything.
I had an idea similar to yours and had a 7x65R/9.3x74R set made up. Instead of getting it made with a bunch of European animals, I ordered it engraved with a Rocky Mountain goat, a Roosevelt elk, and a Columbia Blacktail deer, i.e. USA animals. The 7x65R barrels are fitted with a removable scope. The 9.3x74R barrels have a flip up "moon" sight. The same theme could be applied to a 300/375 set.
