Is it permissible to take medication to induce menstruation if its delay causes embarrassment, knowing that doctors have confirmed there is no medical problem?

There is no harm in taking medication to induce menstruation, provided that the intention is not to avoid acts of worship such as prayer, as Sheikh Ibn Uthaymeen mentioned. It appears that you do not intend to circumvent prayer, so proceed with the treatment if it is beneficial for you and poses no harm, according to your doctor's advice.