L.R.
Did the urine test include a test of his blood sugar level? I assume it did -- if it did not, you need a new and better pediatrician. While this VERY likely is stress-related, the doctor should have checked his blood sugar as a matter of routine because frequent urination (and excessive thirst) is a sign of diabetes. (Even if there is no diabetes in your family your child can still have it-- some parents mistakenly think that if there is no diabetes in anyone else their kid can't have it.) I'd first call the pediatrician's office and ask if they tested blood sugar level and what it was (do not put up with their saying "Oh, it was a bit high but nothing to worry about -- get the specific numbers and ask them for the baseline numbers to which that "little bit high" was compared).
Again, this sounds more like stress, attention-seeking etc. than diabetes-- but if he also is constantly thirsty you definitely would want to get a better blood test done. Just rule it out medically.
Then look at what other stresses your son may have besides the bully -- is this his first time in preschool so it's all new to him? Did he change preschools and misses the old one? What has the preschool done to help him be more comfortable? Does he tend to be immature for his age in other settings, indicating that maybe he's just not ready for preschool?
Has there been any change at home (strife in the household, new sibling, or sibling/parent who has been ill and gotten a ton of attention, grandparent who is ill so a parent is gone a lot caring for that person, parent who is traveling a lot for work, anything)? Even what seem like tiny, insignificant changes to an adult can rock a young child's world, so do not dismiss anything at this point.
Be open to the idea that he might benefit from seeing a counselor who specializes in working with children his age. It's interesting that the doctor said to just ignore it -- did the doctor even bother to ask about stresses in his life or to help you figure it out?? Is the preschool just telling you, "He needs to stop this" or are the preschool teacher and the director trying to help you, recommending resources or counselors, etc.? In other words -- are these professionals trying to help your son or just telling you, ignore it and/or fix it? Be assertive with them and tell them you need some ideas and resources, not just demands to make him stop. You can't. You cannot control another person's toileting - which is exactly why toilet issues are such a struggle; kids KNOW parents can't make them pee or make them stop peeing. But he sounds like there is something going on other than some toilet training power struggle.
Get the medical details, be assertive about checking his sugar, and then be assertive about getting help. Good luck.