Hi Laura ,
Thanks for your post.
You can achieve what you want with ATUM Product Levels.
If I understood your explanation correctly, you can create two BOM products (gold and silver backing), set their stock, and link each one to the variations (pearls A, B, and C for silver rings and the same for gold).
Each time a ring is sold, the linked BOM stock will be updated and also will be updated the BOM's associated products' stock.
You can access all the "Product Levels" documentation in this URL.
Hope this helps,
José Andrés