Expansion depends on velocity. And face it, you're just not going to get much velocity with a slingshot (compared to firearms). If I were experimenting with this, I'd work on making your slingshot ammo tumble rather than expand. For a tumbling design, you don't want spherical ammo (a round ball). Because a round ball that is tumbling (spinning) is still just a round ball.
Take for example two common rounds used in armed conflicts. NATO countries use 5.56x45 (almost, but not exactly, the same as .223 Remington). Eastern bloc countries use 5.45x39, which is similar to NATO's 5.56x45 ballistically. Both rounds can be quite devistating up close - the 5.56 by expansion and the 5.45 by tumbling. (Neither one is worth a darn for penetration however, but that's another discussion). Different designs. The 5.45 bullet looks more like a little spear than a traditional bullet. So when it tumbles, it makes a mess. Neither round is great at distance, but the 5.45 has an edge over the 5.56 there. Because the 5.56 has lost enough velocity so it does not expand, and you end up with this little bitty hole bored cleanly into your target. The 5.45 has lost velocity at distance too, but since it's design is to tumble, not expand, it does a lot more damage at distance since it will still tumble even at decreased velocity. (However, it is harder to stabilize the 5.45 bullet due to it's tumbling design. You don't want it to tumble BEFORE it hits the target, but its design wants to.)
I would think of slingshot ammo in the same light as greatly reduced velocity firearm ammo. It will be too slow to expand, so go for the tumbling aspect instead. Also look at slingshot ammo in the same light as handgun rounds. In NATO countries there has always been split opinion on "light and fast" (e.g., hot 9mm +p ammo) vs. "slow and heavy (e.g., .45acp) The 9mm will either expand and make a nice big wound, or it won't and will leave only a 9mm hole. The .45acp will always create at least a .45 caliber hole, even if it doesn't expand (and non-expansion does happen more often, since it's a relatively slow projectile). Because of the low velocity of a slingshot, I would want the bigger, heavier projectile over any smaller, lighter, faster, expandable (maybe? probably not) projectile every time. If that big heavy thing happens to tumble too, well, that's a bonus!
My personal opinion is that at the low velocities and limited range of slingshot ammo, an expanding/tumbling design probably won't add much, if anything, to the terminal ballistics. And it will probably lower speed and really trash accuracy. I think bigger, heavier conventional ammo and stronger bands will give much more bang for the buck. Just my opinion (based on zero knowledge of slingshot ammo, but at least SOME knowledge of firearm ammo).