Our daughter was just over a year when we went to the pediatric opthalmologist to have her tear duct opened. It's an outpatient surgery at over a year, but can be done in the office under a year. I think I'd probably insist that your doctor refer you to a good one in your area, to see if it's something that will clear up on it's own (ours didn't) and have a chance of doing the in-office procedure versus waiting while he goes under anesthisia, has the surgery, goes through recovery, then comes back to you. It's a very routine procedure, and very safe, but it makes for a long four hours while you wait!
Best wishes and God Bless.