Is prayer invalidated for one who goes back to the opening supplication after starting to recite al-Faatihah?
Question

I forgot the opening supplication (du‘aa’ al-istiftaah) and started to recite al-Faatihah, then I remembered it, so I went back to recite it. Is the prayer invalidated by analogy with the one who goes back to the first tashahhud after forgetting it?
Praise be to Allah.The one who omits an obligatory part or Sunnah action in the prayer, then starts the essential part that comes after it
The basic principle is that if the worshipper omits an obligatory part or Sunnah action in the prayer and starts the essential part that comes after it, he should not go back to that obligatory part or Sunnah action.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
if he remembers it – the obligatory part – after he has moved on to the essential part (posture) that follows it, it is waived and he should not go back to it; rather he should continue with his prayer and do the prostration of forgetfulness before saying the salaam."(Majmoo‘ Fataawa wa Rasaa’il ash-Shaykh Ibn “Uthaymeen  14/98).
Should he go back to the opening supplication after having started to recite al-Faatihah?
Hence if the worshipper has begun the essential part of reciting al-Faatihah, he should not go back to the opening supplication.
Ibn Qudaamah (may Allah have mercy on him) said:
If he forgets the opening supplication, or he omits it deliberately and starts to recite the isti‘aadhah (“A‘oodhu Billahi min ash-shaytaan ir-rajeem”), he should not go back to [the opening supplication], because it is a Sunnah the time for which has passed. Similarly, if he forgets to recite the isti‘aadhah and begins to recite [al-Faatihah], he should not go back to it, for the same reason."(Al-Mughni  2/145).
If he goes back in the manner mentioned in the question, his prayer is not rendered invalid, because it does not change the outward appearance of the prayer.
It says in Haashiyat al-Jamal, a book of Shaafa‘i fiqh (1/456):
If he interrupts al-Faatihah to recite Qunoot or the opening supplication deliberately and knowingly, that does not invalidate his prayer, because it does not change the outward appearance of the prayer, even though it involves interrupting something that is obligatory for the sake of doing something that is supererogatory. End quote.
As for the analogy with the invalidation of the prayer of one who moves back from standing or reciting al-Faatihah to the posture for reciting the tashahhud [i.e., sitting], this analogy is not valid, because actions are stronger than words. Hence the prayer is rendered invalid for one who deliberately repeats bowing or prostration, but it is not rendered invalid for one who deliberately repeats al-Faatihah.
The scholars stated that with regard to the words that are prescribed in the prayer, the prayer is not rendered invalid if they are repeated or recited in the wrong place, except the salaam; the one who recites the salaam before completing his prayer, knowingly and deliberately, has invalidated his prayer.
Saying extra words in the prayer, and the ruling thereon
Ibn Qudaamah (may Allah have mercy on him) said with regard to saying extra words in the prayer:
… The second example, which is saying extra words, falls into two categories:
The first category is that which renders the prayer invalid if done deliberately, such as greeting people or saying words that are part of regular human speech.
The second category is that which does not render the prayer invalid if done deliberately. This is of two types, the first of which is saying words of dhikr that are prescribed in the prayer in the wrong place in the prayer, such as reciting Qur’an when in the posture of bowing and prostrating, or reciting the tashahhud when standing, or sending blessings upon the Prophet (blessings and peace of Allah be on him) in the first tashahhud, or reciting another soorah in the last two rak‘ahs of a four-rak‘ah prayer or in the last rak‘ah of Maghrib, and so on.
The second type is reciting a dhikr or supplication that is not mentioned in any religious text, such as saying “Aameen Rabb al-‘aalameen” or saying in the takbeer “Allahu akbaru kabeeran” and the like."(Al-Mughni  2/426).
Al-Bahooti said in ar-Rawd al-Murbi‘ (2/101):
It is makrooh (disliked) to repeat al-Faatihah. End quote.
Ibn al-Qaasim said, commenting on that:
The difference between an essential part that consists of words and one that consists of movement is that repeating words does not change the appearance of the prayer. End quote.
In other words, repeating an essential part that consists of words does not render the prayer invalid, because it does not change the appearance of the prayer, in contrast to adding an essential part that consists of movement, which does invalidate the prayer, because it changes the appearance of the prayer.
And Allah knows best.