It sounds like you are providing a job description for a Paradox gun. Holland & Holland made them for many years and they make them today. Most of the older ones had 2-1/2" chambers but there were a few made with 2-3/4". The guns made since 2003 all have chambers of at least, and perhaps only, 2-3/4".

The paradox gun is a special kind of ball-and-shot gun with a rifled choke. You shoot standard shotgun shells in it and the patterns are generally from Improved Cylinder to Light Modified (in my experience). When you want to shoot bigger animals you flip up a rifle sight and then, instead of shotgun shells, you insert Paradox cartridges.

I have owned a few Paradox guns and shot some others. I own a couple now, a 1887 with hammers and a 2011 round action hammerless. They are very versatile. I had a 1911 Paradox gun that I used for 90% of all my shooting. I shot sporting clays, grouse, chukar (partridges), pheasant, one bobcat, and two deer. I also took it to Argentina where I used it to shoot doves and a collared peccary. I traded the 1911 gun in on the new one. I will be using the new Paradox gun next month on a bear hunt in Idaho.

Other people who frequent this forum have used Paradox guns on birds and big game in Africa. I have been told that in days past, it was the perfect gun for British officers stationed in India.