Anesthesia has risks so doctors avoid general anesthesia when ever possible. And the use of anesthesia is very expensive.

If you are concerned about the Lasik procedure, talk to the specialist and ask for an oral medication to help you be calm. There are several that really work.

I’ve had lasik, and I have had the same problem as you. I doubt you can be anesthetized (though there is the possibility), but I will tell you right now that it really isn’t that bad. In fact, it was very fast and really painless. When I went in, they just told me to keep looking at the red light. I decided to focus on doing exactly that no matter what, and I did, but “no matter what” didn’t happen—it was much better than I had expected. There is one moment where the red light goes fuzzy, but they have it where you can’t blink or anything and that moment of fuzziness is the only part—it’s interesting more than it is scary. By far the worst part was the anticipation of it. My vision is now even better than perfect vision, and it’s been that way ever since I had it about five years ago. And back then, I was what the doctor called “20-blind.” I could see perfect close, but hardly anything a few feet in front of me. My brother, who had much better eyesight than me, also had the lasik surgery and he hasn’t regretted it, either.
One thing I would highly recommend, though, is to get a good eye doctor (don’t just do the cheap thing on the adds unless you know that doctor is very good). They’re your eyes and are worth the extra money (it would probably be cheaper than bein anesthetized anyway). I made sure I got the best eye doctor I could, and I’ve never regretted that decision. Make sure you tell the doctor your problem. I’m sure he could reassure you or let you know your options, etc.
Good luck with your decision.