Sorry for the delay in reporting the outcome of the hydrogen peroxide soaking... I developed a head cold and couldn't smell anything. But now, I'm better and it appears the pearl is better too. It's completely dried out and odor-free. Don't know what would happen should it ever get wet again though...
It's not the most beautiful, but I don't have the heart to throw it out. So I'll just wear it for awhile and then maybe put it on a silver chain for my 6 year old. She's always wanting to wear whatever I have, so this would be a good compromise!