Hi Reemy,
This does not sound like a good idea to me.
Even under local anesthesia there are medications injected into the body. These typically include lidocaine (which is like novocaine) and epinephrine (which is another name for adrenaline). Sometimes some other medications are used, such as oral Valium and/or a muscle relaxer. And afterwards, you may need a pain reliever, including some that are synthetic narcotics (such as Vicodin or Percocet).
NONE of those things are good for your baby! I'm a big fan of enabling breast-feeding - I try to adapt my surgical procedures (such as breast lifts or reductions, or implants) keeping the potential for future breast-feeding in mind. So having this procedure while you're breast feeding just does not sound like something that would be helpful to your baby....I doubt you'd want to administer any of these medications directly to your baby!
The next question is how long you would need to wait before resuming breast-feeding, and I would think AT LEAST a few days after you're received your last medication. You should check with your OB and your medical doctor to get the specifics on this.
So if you're really motivated to have the procedure now, you could always pump (and discard) during the period in question, as long as you are prepared to hold off resuming the breast-feeding until all your docs are comfortable with that amount of time. It seems a lot to do during an important time, but I understand the desire to get things done in a specific time period. As long as you are willing to stay within the advised guidelines, it is a decision that sounds do-able.
I hope that this helps, and good luck,
Dr. E