Whichever way you go, a decent 20 gauge SxS should weigh right about 6 pounds. many of the very inexpensive ones weight closer to 7 or more.
The gun needs to FIT - cast seems to play more of a role with SxS than almost any other type of shotgun, along with drop, pitch, etc. - but when you find one that does fit - it will swing like it has its own radar to the target.........