It's completely legitimate to go vegan for animal rights reasons. But lactose intolerance does not, and cannot, cause headaches. It is a specific disorder affecting the lower intestine and causes diarrhea and gas pain. I am afraid that your posting may be describing some other disorder (or it may be effectively another example of colpo d'aria).
*Edit: for those of you who don't know, colpo d'aria is a made-up disease used by Italian mothers to explain why their kids have to be bundled up ridiculously in order to go out in the cold: the closest term in English would be "catching your death". Italians are extraordinarily superstitious and hyperchondriacal about diseases and disorders. Frauds run rampant and homeopathy is openly advertised in drug stores as if it's real.
*Edit: for those of you who don't know, colpo d'aria is a made-up disease used by Italian mothers to explain why their kids have to be bundled up ridiculously in order to go out in the cold: the closest term in English would be "catching your death". Italians are extraordinarily superstitious and hyperchondriacal about diseases and disorders. Frauds run rampant and homeopathy is openly advertised in drug stores as if it's real.