Hello and welcome Lisa.
Based on our experience, I think I would listen to the surgeon. My husband's primary as well as a cancerous lymph node were not picked up by the PET scan.

The neck dissection sounds alot worse than it is, for most people. My husband passed out in the surgeon's office when they first described the surgery to him. But, chances are, your dad will think the radiation and chemo were alot worse.

Perhaps a second opinion would help with the decision.

Best,
Anita