All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
If the doubts that you have about missing the transition takbeer (saying Allaahu Akbar) or mispronouncing consist of mere waswaas (obsessive whispering), then you should totally ignore such doubts and carry on with your prayer without repeating the takbeer. When a person often has such doubts about his prayer, then he should disregard them and not repeat what he doubts. However, if you have such doubts occasionally and not on a daily basis, then you should say the takbeer that you doubt that you have missed because the original state is that you did not say it.
In this case, you did right by correcting the mistake and repeating the transitional takbeer. The rule in this regard is that when the praying person doubts missing any of the obligations of the prayer, he should perform it because the basic principle in this case is that he did not perform that obligation.
As for doubting the mispronunciation of the takbeer, you should disregard such doubts because the basic principle in this case is to act upon what is certain, which is that you have pronounced the takbeer correctly. Mere doubts are not adequate to go against the basic principle in this regard. The Maaliki scholar An-Nafraawi wrote, "That could be because, in principle, one should act upon what is established with certainty."
As for your statement, "I was aware that I was doing more when I repeated the prayer or phrase a third time," if you mean that you repeated the takbeer to correct your mispronunciation, then you have already known the relevant ruling. However, if you mean that you deliberately recited an extra takbeer without a need, then you should have not done so. However, this does not invalidate your prayer because the scholars underlined that the prayer is invalidated by deliberately adding an extra physical act among the pillars of the prayer and not a verbal one. The Maaliki scholar An-Nafraawi wrote, "It is deduced from the statement of Khaleel: 'Deliberately doing an extra physical pillar of the prayer, such as a prostration, invalidates the prayer' that adding an extra verbal pillar does not invalidate it." [Al-Fawaakih Ad-Dawaani]
The Hanbali scholar Ar-Ruhaybaani wrote, "The prayer is also invalidated by deliberately adding an extra physical pillar of the prayer such as a rukooʻ (bowing) or sujood (prostration)." [Mataalib Uli An-Nuha]
The majority of the scholars held that reciting the transitional takbeer is recommended (not obligatory); hence, the prayer is not invalidated by missing it or mispronouncing it. The Hanbalis, on the other hand, maintained that deliberately missing the transitional takbeer invalidates the prayer.
Hence, based on the information provided in the question, it seems that your prayer is valid in all cases, and you do not need to repeat it.
Allaah knows best.