I Have A 5year Old Girl Who Has A Cyst

I have a 5year old girl who has a cyst on her left eye that she has had for. Wile and her pediTrician said it is a cyst and when yu touch it it moves freely but she said we should leave it alone but we want to have it removed as it does stand out. The prob is that I was told that she would have to be intubated to remove the cyst as she is a child but I don't want to have her incubated because she has vocal paralysis on one of her vocal chords so I don't want her voice compromised any further with the intubation. What kind of doctor would do this surgery if we pursue this? Who should we see to evaluate her cyst beside her pediatrician? Why can't she have the cyst removed without general anesthesia? Please let me know