I understand what you're saying--my husband is the same way; however, they can keep giving you shots until they're done with the surgery. I had to have extra shots to deaden my mouth one time during the middle of a tooth extraction. Just be sure they know that it wears off fast and then maybe have an agreed upon hand signal to tell them you need another dose.
Also, I believe they have different strengths of these. The last time hubby and I were at the doctor and we explained that locals didn't work for him, they showed us a bottle of stuff that they said they'd had to use for others like him--and it worked on them. It's worth investigating!