I vote for 8x57IRS or 9.3x74R, Both are great calibers and are available in US now. Also both fit very well in 16 ga action. There is nothing wrong with WBLDon's choices( except maybe 45-70 needs 12 ga action), these are just mine. Note they are all rimmed and to quote my late brother in law" a double barrel rifle belongs to be chambered for a rimmed cartridge".
Mike
Both of these are excellent cartridges for a double rifle, as well as the list of American rimed cartridges before your post. Click on the link below to watch a video of a German lady hunter make a one shot kill with a 8x57irs during deer stalking in Scotland. This is an excellent professional produced hunting video seldom seen today.
https://www.youtube.com/watch?v=ttMKUzReVKU
