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!