Which doctor should you see to remove a filiform wart, an ophthalmologist or a surgeon?

Above my left eye on the upper eyelid I have a filiform wart which is about 1cm long and when I turn it comes out in front of my eye. Yesterday I spoke to the surgeon whether he could take it away. He told me that because it is directly above the eye he cannot give me anesthetic and that he would have to cut it out and put in one stitch. It seemed to me that he was NOT so sure whether he could do it directly above my left eye and he had to ask the nurse if she could do it. I also somehow have a bad feeling that he might accidentally injure the nerve in my eyelid which is causing me problems closing my eyes. Now my question is whether I should have the wart removed by the ophthalmologist who is treating me, for safety reasons.