I tried asking my doctor and the GP practice if they would do that for me and the answer was 'No'.
However, what I did end up with was a prescription for the large tubes of EMLA cream — which contains lidocaine — and combined with a codeine-based painkiller (tablet) that did the job for me, including for the dozen or more genital electrolysis sessions I had prior to surgery.
That's not to say that there was still pain and some gritting of teeth, but it was at least bearable. And also, a level of pain which I became accustomed to over time.
With the EMLA, I found that repeated applications over a period of about 90 minutes before the session seemed to work the best.
Sorry if that's not exactly what you asked, but I thought I'd just put that out there as an alternative if your dentist is as helpful as my doctor was.