Most of the larger, longer running international schools in Bali have some form of counselor on staff, often called a wellbeing coordinator, pastoral care lead, or school counselor depending on the school's structure. The exact setup differs a lot between campuses. A big K-12 school might have a dedicated counselor for the primary years and another for secondary, while a smaller school newer to the island's growing list of international schools in Bali may fold counseling duties into a teacher's existing role.
Ask directly rather than assume the role exists. Some schools that call themselves international do not have a trained mental health professional on site, they rely on a teacher who has taken on pastoral care alongside a full teaching load. Others have a proper counselor with a psychology or social work background who meets students one on one, runs small group sessions, and works with parents when something is going on at home. Bring this up during the admissions process and ask to meet the counselor directly, then ask how they handle anxiety, bullying, or a child struggling with the move to Bali.
If a school does not have anyone in house, your child is not without options. Canggu, Ubud, and Sanur all have private therapists and psychologists who work with expat families and children, and several offer sessions over video call as well. Check what your private international health insurance covers for therapy and mental health visits before you enroll rather than after.
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