The risk of infection is probably much larger and the piercing might take twice as long to heal just becuase your body is putting so much energy into making milk. There is always a larger risk of infection with mouth piercings anyway but if I were you I would wait it out.

I highly doubt they would pierce you as you have to sign an agreement stating that you are not sick, pregnant or brestfeeding. If you did get an infection and it did spread to other body parts it could get to your baby via milk so I wouldn't risk it, but it is your call.