Rear suspension travel

The kit I have in mind is an anti-roll bar, like you would see on a 4 link. I’m not convinced the issue is side to side movement but more of a body roll. How much can our leaf spring suspensions move side to side?
Dukeboy has it right... the leaf suspension is going to move sideways quite a bit during cornering. That is at least half of the problem, if not most of it. How much clearance is there between the tire and body now, when sitting still? Push repeatedly sideways on the rear bumper at a frequency that matches the natural sideways frequency of the rear (like pushing a swing higher and higher) and you will get some idea of how much it moves. An anti-sway bar is not going to fix ANY of this movement.

Attacking this with just a heavy rear anti-sway bar does effect the relative roll resistance of the rear versus the front, and in extreme cases, then the rear will want to break loose sooner (oversteer); on a slick surface, like after a light rainshower, it might 'catch you out'. So beware.....

Sideways movement for a leaf suspension is attacked typically with panhard/watts links or mono-leafs. The advantage of a panhard is that you can make the link connection point variable and thus raise or lower the rear roll center, which is an advantage in adjusting handling. That is what the circle track boys are doing when they 'adjust the track bar'. (You DO have to be aware of this factor when installing a panhard rod, and make sure you do not mess up the rear roll center!)