I like how you are asking about a "small" double then go right to the 470NE and 500NE.
A double in 375H&H makes a ton of sense, but the lack of a rimmed case is a minor downfall when it comes to extraction/ejection. Moving up slightly to the 450/400NE (3 inch) is the next practical step that also has a rimmed case and great history. Another option would be the 500/416NE that is made to replicate the 416RIGBY ballistics in a double rifle rimmed case.
As for optics or not... I'm firmly in the "have magnified optics" camp, meaning I like something with an illuminated reticle, true 1x at low power and up to 6x or 8x magnification at the upper end. This would set in a quick disconnect (QD) mount. My preference is the Swaro Z8i line of scopes. At a minimum, I would want a very rugged and reliable red dot like the Trijicon RMR-HD or SRO, but that will limit my range for taking game to about 100 yards or less...mostly less.
The age and condition of your eyes is also a consideration. No one likes to admit it, but eyesight NEVER gets better with age. You can have surgery to make repairs but it will always be going downhill. As I have aged, I've found that I need more magnification than I did as a youngster.
Mounting on a double rifle. You are going to need a special mount to make this work. Depends on the rifle you buy and what optic you plan on using. For instance...buy a Blaser S2 in 500/416NE and you can put nearly any optic you wish in a Blaser QD mount. That Sabatti you are looking at in 500NE is going to require some custom fitting (meaning more money) to get a scope or red dot mounted.
Another consideration is regulation. The easiest way to do this is reloading yourself and testing the accuracy of each barrel. If you don't reload or know someone who does, it adds time to the process as well as money to get it done. Many used double rifles are sold with dies for this very reason.
IMO - If you REALLY want a double, get it...but understand that there are limitations to that platform vs a bolt action rifle. Not to mention the significant gap in price between the two of them. I would rather spend about $6K to $8K on a semi-custom bolt action rifle/scope combo that was DG proven reliable and capable of taking game out to 200 yards easily. Then put the additional double rifle money towards another animal on safari. Do your first trip with a bolt action DG legal rifle caliber and go from there.
That's what I was thinking when the title was a "small double rifle".
Although I was going to say a 6.5x57R or even a 7x65R would be nice.