A drug applied topically. Can be the same drug or other drug.
The problem is that religious thinking is not compatible with modern medicine.
Anesthesia inhibits thermoregulation on babies. It is common for core temperature to drop by 1 to 2 degrees Celsius in the first hour of anesthesia. Sick babies can became hypothermic easily and struggle to keep themselves warm. Add anesthesia to that mix and you could be reducing the survival rate.
Some researchers also claim that anesthetic exposure can cause toxicity and neuronal apoptosis (death of neurons) in the developing brain of a baby, and this has the potential to became a long term problem in the later nerve development.
This is a problem particularly for premature infants. This babies sometimes need to pass by multiple surgical procedures with multiple anesthesia. Premature birth is a main cause of death in children, so they have yet a lot of things to address even without the toxics.
Avoiding punctual pain at any cost shouldn't never be the main (and much less the only) objective here. And adding a new risk to the medical procedure, just for ideological reasons, is very stupid.
It's also very well documented so no it's not about social warriors