I'm not sure that I think that an HIV+ surgeon must disclose their status to their patients... Patients certainly don't have to make a reciprocal disclosure. I don't think that the physician loses the right to privacy of their personal health information, nor do I think that they deserve to be stigmatized by limiting their practice options. It would be different if there weren't layers of precautions that could be taken. Especially if the surgeon is on anti-retrovirals, they may have such a low viral load that they would pose little to no risk to a patient even if a needlestick occurred.
That said, knowing how common needlestick injuries are, and having experienced several myself, despite being personally very cautious with sharps... if I knew that I had an infectious bloodborne pathogen on board, I would be very unlikely to go into surgery. I've been stabbed with an injection needle that a resident had left unguarded in a place that it shouldn't have been. I have stepped on a very large bloody suture needle that went clean through the bottom of my puncture resistant shoe and into my foot. Both of those couldn't have been prevented by me being more vigilant about how I handled sharps... though in both cases, there was no risk of my blood contaminating a patient. I was not a surgeon.
I have seen a surgeon, who is very disciplined with sharps, stabbed in the hand by a careless scrub tech during a spinal fusion. His blood absolutely hit the open wound and contaminated the patient. No amount of increased caution can protect against an idiot elsewhere on the team.
I would simply not be able to live with the idea that I put a patient at risk of contracting an incurable, potentially life-limiting infection from me. I wouldn't be able to sleep at night worrying about exposures that I may not have noticed. Maybe there is a nonsurgical field OP might be interested in? You gotta follow your own conscience, but if it were me, that is what I'd pursue.