I would not recommend having this surgery performed with only local anesthesia. It would be difficult for you as well as the surgeon and has the potential to impact your results. The real question is why you feel local anesthesia is preferable to you over general. I would advise you to meet with a Board Certified plastic surgeon who can talk to you about the risks/benefits of this surgery as well as address your concerns about anesthesia.
I have performed this technique under local anaesthetic and sedation on a couple of occasions for patients who were unable to have general anaesthesia. In both scenarios it was a very modest lift. I found that even with anaesthetic blocks that I needed to infiltrate quite a lot of anaesthetic solution locally. I felt this distorted the tissues and made it more difficult to perform the technique.I personally would recommend a general anaesthetic for this procedure unless there is a medical reason that you need to avoid general anaesthetic.