I thought of doing that but i also dabble in robotics so the electrical component of the project was a natural fit. You are correct though this ain’t a sniper rife so tight grouping’s are out. :)
One US source of tungsten shot. https://www.ballisticproducts.com/mobile/Tungsten-180-Shot-9-2-lbs_jar/productinfo/TSS09/
Not really financially viable
Just spit balling here but I could encapsulate lead shot in West System epoxy. If the unit broke the internal weight package would be a solid unit and bonded to the plastic cover. The epoxy would also maintain unit integrity.
I could also use a length of solid steel cylinder. It will take care of the voids that would be created by using bbs. I don’t know in what $ range your suggestion would fall or its available form factors and availability in general considering today’s supply chain shortages.