Aerodynamics are not gentle to spherical objects at supersonic speed. You can ramp up the initial speed, but shot pellets are going to decelerate rapidly. And the supersonic shock waves/transonic transition is going to wreak havoc on the pattern.
Here is an article about the ballistics of spherical objects (including shot pellets and muzzle-loader balls) at supersonic speeds.
https://www.sciencedirect.com/science/article/pii/S2214914717301459#fig1
Here's a drag curve graph from that article:
Notice how high the drag gets and stays? Compare that with drag curves for rifle-bullet-shaped projectiles:
Note that the scales on the left are not the same. The bullet scale show tops out at 0.7 drag, while the spherical object chart runs up to 1.2.
What does this mean? Extremely rapid decay of speed for spherical objects driven to supersonic speeds. This chart purports to show what happens to #6 shot launched at 1400 fps:
Notice that the pellets are back down below 1000 fps at 20 yards. They are slamming on the brakes!
A decidedly-subsonic example may help illustrate. Have you ever thrown a ping-pong ball? You can throw it as hard as you want, but it sort of has a built in speed-limit... you can exceed that velocity, but it is going to slow down
very hard until it gets back to that speed. You can have
Mike Foltynewicz throwing a ping pong ball in a game of catch with you, and if you are standing more than a few feet away, the balls he throws to you and the ones you throw to him will be
caught at close to the same velocity.... even if he's starting his out 40 mph harder.
So, in answer to your question, there is no
internal ballistics reason in terms of gun design that one cannot build a gun of shoulder-able weight that would throw pellets at 2200 fps. However, there is no
external ballistics reason to want to do that. You would
dramatically increase the recoil and you wouldn't really get pellets flying much faster... and your pattern would probably be much worse at long distances (which would presumably be the place where you'd most want to use such a gun) than a conventional-velocity shotgun.
Note that this is why anti-aircraft guns are not big shotguns... they are firing single projectiles (rapidly, but only one at a time) that get to the vicinity of the target and
then explode.