My daughter has 4 hemangiomas - 2 deep (1 parotid - on cheek near ear, and one on her fingertip), 1 compound on her shoulder and 1 superficial on her chest. We sought treatment because of the parotid hemangioma which was growing rapidly and was 6 cm across at 7 weeks of age.
Your daughter is beautiful! Also, your feelings are very normal! I don't mean to make you feel worse, but your daughter's h looks to be getting dangerously close to her eye. You didn't mention where you are from, but I'm wondering if you have sought the opinion of a specialist, or who it was that was going to do the surgery? I would contact the experts on this site and seek their opinion - send them pictures. Depending on the experts opinion, I would also seek out someone local who is willing to prescribe Propranolol for your daughter. Have you done any research on this drug? Do a search on this site for "propranolol" and you can read a number of parent's stories. My daughter is currently on it for her parotid and and it worked wonders. Dr. Waner recommended it for my daughter. In just a few weeks it went from a lesion that was disfiguring her face to not even noticable if you don't know she has it. Propranolol is not without risks, and you should do your research, but with proper monitoring, it can be a wonderful treatment. I think, that with possible eye involvement, it may be a very good option, and I think it may be the option the experts point you towards.
If you have any questions, please feel free to e-mail me directly.
My daughter is also 4 months old by the way!