Yes—a shoot-through white umbrella can work, but for small metal products there isn’t one single “best” diffuser. The bigger issue is controlling reflections safely.

Do not tape printer paper to hot photoflood lights; that can be a fire risk. If your current diffusion reduces light, compensate with a tripod and longer shutter speeds rather than removing diffusion.

For metal subjects, large diffused sources usually work best. A white shoot-through umbrella is one option. Another often better option for product work is bouncing the light into large white foam boards, then adjusting the board angles to shape the reflections. This can give smoother highlights and better definition on brushed metal or engraved surfaces. Adding thin black strips to the white boards can also help break up flat reflections and define edges.

If possible, consider replacing hot photofloods with cooler continuous lights, since modern cameras make white balance correction easy and cooler lights are safer around diffusion materials.

So: use safe diffusion, keep the light source large relative to the product, and control reflections with white boards rather than relying only on a specific umbrella.