Unfortunately my man, your question, here, has two problems.
1st, the question itself, and it's title. As you've seen, people don't always read things as you think you wrote them, sometimes they don't read at all, sometimes they scan the o.p. and skip the replies just to get their 1.5 cents in. Thus, you will have people expound on the hunting and target versatility of their favored smoothbore. Didn't take but 5 or so replies after your reminder here for another to crop up.
Second, the breadth of the question.
You've narrowed you parameters in subsequent replies and that helps, but, your time frame is just too vast.
Their really just isn't one gun that covers the whole thing. Maybe, maybe, you could get by with two,,,, with some really creative backstories,,,,,,,, maybe.
Having this be the case encourages overspeculation and people creating their own truth or finding creative ways to deny historical evidence (much like certain elements of our society today) in order to support their gun or style of preference.
Honestly, the most creative I can get with backstory and "justification" is far stretched and would need a greater budget. A greater budget than two India guns most likely.
Best of luck on your quest.