Flat leather slingshot round head approach

Can't change it, but try mounting the flat leather to a regular garden leather slingshot.

I've seen others do this

For a ballcock slingshot, wrap the end of the flat leather around the ballcock, then tie a thin leather strap to keep the ballcock from falling out.

Traditional big-eye slingshot, with a thin wire hanging from the slingshot, and a flat leather strap and thin wire to hold it in place.