And a rear sway bar will not stop the axle from shifting side to side under the car. Depending on how it's mounted it might add a small amount of resistance, but we're talking a very small amount, just what the bushings are giving you when you're pulling the end links sideways. Definitely not the reason you'd want to add one.
And looking at the sway bar kit the OP is considering, it won't stop any sideways movement since the vertical links have heim joints at the ends. It'll just reduce the body roll.
