I found a recent paper on the physics of the sling from students of the university of Granada: https://doi.org/10.1119/5.0128128

It can be accessed here: https://digibug.ugr.es/bitstream/handle/10481/89137/Sling_Shooting_Accepted_Manu...

They fixed a sling to a power drill with the release cord being held by an actuated clasp. I doubt that this can provide insights to the mechanics of the throw, but their overall setup is quite accessible and works well. Might be interesting to look into