The immune system does its job fine on measles for kids and teens that eat that way everyday in the parts of the world where measles vaccination is widespread. It also does its job fine on the kids who have been vaccinated in areas where vaccination is not widespread.

Also consider before there was measles vaccination. Large deadly outbreaks were common all over the world in areas with vastly different diets and lifestyles.

This suggests that diet is unlikely to be a significant factor.