One way to solve these problems is to have a gun that can fire full powered cartridges for long range or penetration and multiple projection cartridges, with say 3 bullets, for short range. These multiple bullet rounds would only have to be accurate and able to penetrate soft body armor out to 1-200 meters, ideally with at least one projectile hitting to point of aim and the others hitting within a 1-2 foot radius. While current cartridges could be used a new one with a long neck would probably be best. Whether it should be semi-auto only or have burst or full auto capability will have to be worked out. This doesn't completely solve the carrying problem since while each multiple projection cartridge equals three regular cartridges fired on burst, the advantage dissappears when slow aimed fire is desired.
The Austrian firm of Hirtenberg made tungsten cored 5.56 bullets in the '70s with excellent penetration.