With due respect to your GP, the diagnosis of neuralgia and your concern about myelin sheath damage and possible "lifelong" pain attacks are not supported by anatomy, science, or laser biophysics.
I'm not sure what you mean by a "thread vein" since that is neither a medical term nor a diagnosis. I presume you may be referring to a lip telangiectasia (a visible small dilated vein), which is very appropriate for laser treatment as they are superficial enough to be reached by the laser energy (close to the surface), appropriate diameter (small), and with a chromophore (hemoglobin) well-absorbed by several laser wavelengths (KTP at 532nm being one of the most common). Other lasers could also be used effectively, but all have similar characteristics if used for treatment of superficial vessels.
You clearly described severe pain, and felt "as if my nerve had been hit." Actually, at the surface of your lip skin, there are only many very sensitive tiny nerve endings, which is a good thing for being aware of food or a gentle kiss, but not so good when a painful stimulus like laser energy is focused there! Imagine using a magnifying glass to focus the sun on a tiny point on your finger tip, another area of many tiny sensitive nerve endings. Anywhere on your lip would have been equally painful--you did not have a nerve branch or larger nerve injured at all by the laser energy. It is anatomically impossible, despite your concern having been (incorrectly) validated by your GP's dismissive and quick-to-blame agreement!
First of all, the infra-orbital nerve branches to the upper lip (labial branches) are too deep to have been reached by the laser energy, which penetrates no more than 0.05-1.0mm below the surface of the skin at vascular-absorbing wavelengths. The same applies to the labial branches of the mental nerves that supply sensation to the lower lip. Furthermore, the formation of a tiny white blister indicates that the target vein was indeed coagulated, absorbing the laser energy (as intended) and causing the appropriate heat damage to create a "tiny white blister" indicative that the treatment went no deeper than the target vein. Little if any energy was "left" to penetrate deeper! Your blister turned into a scab because it was allowed to dry out (I would have suggested Bacitracin for the 3-6 days until healed), but even a tiny scab should heal as uneventfully as a pimple!
Neuralgia is defined as "paroxysmal pain which extends along the course of one or more nerves." You do not describe the pain you have (or had), but even initial persistent discomfort in the treatment area should quickly resolve without any concern or long-term sequelae. Your nerves' myelin sheaths were not damaged by the laser energy (see depth of penetration at vascular-absorbing wavelengths above). Any (very) localized tissue damage (localized because of the collimated or focused laser beam) would quickly heal with very low likelihood of leaving a permanent scar, and high likelihood of eliminating the offending vessel!
In my humble opinion, your GP should have reassured you rather than fuel your anxiety! You need no medication at all for your laser treatment--if anything else is going on, it is independent of, and unrelated to, your vein removal! Relax and be well!