Just about any band will shoot a BB. Given the light weight of the projectile, you have almost total freedom to play with all the parameters.
You can pretty much decide your projectile velocity, draw strength, draw length and velocity sensitivity to heavier projectiles as you wish.
The trick to shooting BBs is to not shoot them too fast. Over 100m/s you may start running into spin effects. Denser BBs suffer less from this.
I personally like Theraband Black 230mm, 18mm at the fork to 12mm at the pouch. 2 bands per side. Lightweight pouch. Shot at near max elongation semi butterfly draw. 4-6mm lead roundball.
It's a superlight draw, good sight picture, negligable drop, good energy, low spin, not too sensitive to ammo weight