What great news! I am so glad you got a confirmation for your diagnosis. Do you have a scan or something that you feel comfortable sharing? That would be very interesting to see how long your styloid is & the position.
As far as external or internal, Isaiah makes some good points. Besides those, I think purely from a recovery standpoint after having my own external surgery, I think it would be harder to deal with an incision in the throat/mouth. Immediately after my surgery for the first several days, I was unable to swallow. I didn’t like that feeling. Now, I just have some minimal First Bite Syndrome when I eat certain foods & I have to deal with pain & numbness in my neck. I think having something done in my throat would be a whole new level of issues & pain tolerance. But that is just my thoughts. It sounds like your doctor normally does the external. Just imagine not having your pain issues & coughing minimized or gone. That would be awesome! So happy you are on this new path towards healing.