Is using medication to induce menstruation in Ramadan, for the purpose of finding an excuse to break the fast and avoid embarrassing family, considered intentional breaking of the fast that necessitates making up for it?

You made a mistake by taking medication to induce menstruation in Ramadan with the intention of breaking the fast. What you mentioned does not permit that, and it is not permissible to induce it for the purpose of breaking the fast in Ramadan. Some jurists have stated that menstruation induced by medication before its usual time does not invalidate prayer or fasting.