Well I had an excisional biopsy and the tissue removed was comparable to the size you mentioned. They said it was done under local BUT I was given IV sedation as well and I never knew a thing. As far as I was concerned I was asleep. When these methods are combined it's like a general anesthesia without the risks. You definitely should have a talk with your surgeon ... he/she can answer all your concerns.
I had a 2.1 cm tumor removed excisionally, followed by a re-exision for clean margins. I can't imagine your doctor would do this under local anesthesia, but I suggest you talk to him before the surgery to get your questions, and your fears, addressed.