I'd be concerned that at the scale of difference proposed, slings and bows are not very different at all. In DBA for instance, LI, LS and LB are all just called "Psiloi", or light skirmishers. I think one if the things that's weird in CC:A is that there are so many different Light Units already, yet no real distinction between general heavy infantry, hoplites, phalanx, and legion.
In some games, Balearic Slingers hit in SWX if battling back. In our games, we let Numidian elite LC hit in SWX in close combat, but they get no ranged fire, to make them a bit more like MC, which is how they were used.
Actually the sling was an excellent weapon for it's time, in no way inferior to the bow. As far as I know, the firing rate is equal, or maybe 1,5/1 for the slings. The effective range distance is also in favour of the slings. However it is not possible to fire while moving, or that will be at the cost of drastically reduced effectiveness. Whoever is interested in history and usage in that simple but deadly weapon may find a lot information and illustrations at:
There are also a lot of manuals for do-it-yourself slings if you are eager to try yourself. I braided myself one.
The problem with the sling is that you have to have a lot of practise to use it effectively (this is why Balearic slingers were such an asset to an ancient army). Another issue is that it is more of a scirmish weapon, than one that you can use for firing intense volleys as a group, thus gradually loosing popularity in the late years.