This is a small-game hunting tool, not a weapon. The pellet threatens only soft-tissues and eyes. Stat it like a strong blow-gun, not a stronger sling.
Regarding the stats, I think that they're too much for a simple weapon with ammunition. You've statted a short-bow with a poorer crit multiplier and shorter range. I'd make it martial, but I think that it's unrealistic as an actually threatening weapon. (See below)
A sling is actually more difficult to use IRL than a bow; you have to load it with a bullet or stone, whirl it without hitting either yourself or an adjacent ally, and release it when it's 'pointing' slightly off of your intended target. Only the idea that 'throwing rocks' is easy, and that the materials are ubiquitous makes it a 'simple' weapon.
Additionally, the sling's action, coupled with the mass of the bullet actually imparts significant momentum to the projectile when hurled. The damage from 2 to 8 Oz. of lead, clay or rock is quite impressive.
A bow shoots as far as it does because the flights on the arrow, coupled with the arrow's shape, make it so much more aerodynamic than a bullet or stone, that it acts much like a glider. However, the momentum of a slung stone is such that it can rival, or even surpass, an arrow for distance.
This pellet bow appears to be at least as difficult to operate as a bow, but must use lighter pellets than a sling. The damage, if verisimilitude is a goal, should be lower and the range shorter than that of a sling, due to the necessarily smaller mass of the projectile, with no increased aerodynamics.
The weapon would be useful for small game or varmints, but much less so versus any large game or armored opposition. Think of the modern 'Wrist Rocket' sling-shot. It's useful versus a rabbit, but you endanger only a person's eyes or soft tissue. A sling, capable of lofting projectiles so much heavier, can break bones and cause significant damage to skulls or internal organs.