Moon girl, two of my four children had this. There are options besides surgery. I'm surprised your doctor didn't tell you about them. One option is for the pediatrician to put silver nitrate on it. This is a painless procedure. The silver nitrate causes the granuloma to quickly dry up and fall off. This method is not always succesful, however and if you try it, make sure you dress your ds in an outfit you don't care about and wear old clothes yourself, because it makes permanent stains and it will get all over you when you pick up your ds after it's applied. This method worked for my dd, but not for my ds.
For ds, after the silver nitrate failed, our pediatrician gave me two other options:1.)surgery and 2.) to let her tie a string around it, tightly, which would cut off the blood supply to the granuloma and cause it to fall off. She assured me that the string method was painless, because the granuloma has no nerve endings. I chose this method because it seemed less invasive. So right then and there, she tied it up (it did NOT hurt ds) and by the next day, the thing had turned black and fallen off. No further problems.
The pediatrician did freak me out by asking if pee ever squirted out of it. I was like, "HELLO! If pee was coming out of my baby's belly button, don't you think I'd have mentioned it?" Apparently, this happens sometimes.
She also said that umbilical granulomas are becoming very common and that she'd read an article which hypothesized that this is because of the way we're instructed to obsessively wipe the umbillical stump with alcohol. Apparently, the alcohol prevents the natural bacteria which should cause the stump to decay from doing its job.