Does my moving away from my mother—at my father's request—to live with my grandmother, who resides far from my parents' home, constitute an act of disobedience towards her, knowing that she does not object to it but sometimes needs my help?

The mother's right is great, and Allah has commanded kindness to her, and given her priority in companionship. Your living with your grandmother at your father's request and your mother's consent is not considered disobedience. However, if you find an opportunity to visit your mother and help her without disrupting your grandmother's affairs, this would be closer to what is right and a way to please everyone.