The Kurgan, you are looking for the impossible. For instance a gun which excels in trap will shoot high and be fairly heavy. While this might also work acceptably for shooting geese from a pit, it will not be any good for deer and will be poor for skeet. Similarly a gun optimized for deer will be horrible on the trap field. The compromises required to fill your specifications mean that you end up with a shotgun that does not do anything particularly well. Even with interchangeable barrels, the stock configuration will limit usefulness in one area or another. Who wants to spend $2K for that?
Fortunately, your budget allows for the purchase of two or more guns. My advice would be to prioritize your shooting wants and buy two shotguns, each one optimized for the type of shooting most important to you. That way you will end up with only a couple of compromises and have satisfactory guns for the things you enjoy the most.
Think of it this way - do you ever ask for a sleek convertible sports car that has 1 G cornering capabilities, but must have 10 inches of ground clearance and the ability to tow 15,000 lbs.? The goals are mutually incompatible, you must buy a sports car and a truck to satisfy all these needs. It is the same thing with the shotgun - no size fits all.
Sharptail