I’m a little confused. Does the doctor want to cut those nerves in conjunction with doing a styloidectomy or just to try to relieve your pain without doing anything about your elongated styloid(s)? It doesn’t seem like a normal part of a styloidectomy procedure. The greater auricular nerve has been mentioned in several posts which you can find by using the magnifying glass search tool in the upper right of this page & typing greater auricular nerve in the search box. One post mentions that the GAN was moved out of the way during surgery but not cut: Surgery with Mr Hughes London this Friday - #309 by PatientD
You can also search for posts discussing cutaneous nerves the same way. Here’s the link to one post that mentions the brachial cutaneous nerve was moved out of the way but not cut during surgery: Surgery Completed! - #10 by tokenegret