Praise be to Allah.
Undoubtedly shyness is one of the greatest of good characteristics, especially in a woman, and especially if she is a young girl at this age.
If your shyness prevented you from speaking frankly about fasting to make up the days that you missed in Ramadan because of your menses, you could have used some kind of double entendre or vague words. So you could have said for example, “I am fasting like everyone else in Shawwaal,” or “Fasting is prescribed in Ramadan and at other times,” and the like.
Your saying, “I am fasting the fast of the patient,”may carry a sound meaning, because this is how fasting is: it is the deed of the patient and fasting is a deed that requires patience, so it was not necessarily a pure lie, but that is on condition that when you said it you had this sound meaning in mind.
Whatever the case, we hope that what you said may be forgiven and overlooked by Allah, because you were overwhelmed by shyness that distracted you and made you unable to think of what to say or to come up with appropriate words.
For more information, please see fatwa no. 27261 about the ruling on using double entendres.
Fasting depends on the intention, not what the fasting person says unintentionally. If you intended to make up missed fasts, then it will be counted according to your intention, and this fast is valid and is sufficient to make up what you missed during your menses, even if you told your father that your fast was naafil or something else. None of that has any impact on the ruling on your fast, because the Prophet (blessings and peace of Allah be upon him) said: “Actions are but by intentions, and each person will have but that which he intended.” Narrated by al-Bukhaari (1) and Muslim (1907).
For more information, please see fatwa no. 109351
Based on that, you do not have to make up these days, or to put yourself through the hardship of the schedule that you mentioned. Rather what is prescribed in your case is to offer whatever naafil fasts you can, without burdening yourself with something that it is not proven that you have to do in the first place.
And Allah knows best.