Is repeating the opening "Allahu Akbar" (Takbirat al-Ihram) after the first intention and before the opening supplication (Du'a al-Istiftah) in a silent prayer, without the knowledge of the congregants, considered to invalidate the prayer, especially given the belief that this action stems from a satanic whisper (waswasah)?

One must continue the prayer as long as the intention for the present prayer was made before the takbir. Any perceived intention for a prayer other than the present one after the takbir is not considered. Your takbir for a second prayer invalidates the first prayer. Some jurists have made an exception for the validity of a follower joining the imam in two situations: if he began praying alone and then joined the congregation, or if the imam and the congregation began the prayer, and then the imam doubted his intention and repeated the takbir. It may indicate that the prayer of the ma'mum (follower) is not invalidated if he begins the ihram before the imam due to an excuse, based on what Ahmad and Abu Dawud narrated from Abu Bakra: that the Messenger of Allah, peace and blessings be upon him, entered the Fajr prayer, then gestured with his hand for them to stay in their places, then he came, and his head was dripping, and he led them in prayer, and it was not mentioned that they repeated the takbir. This can be used as evidence for the validity of a follower joining an imam who is in a state of ritual impurity if the follower is unaware of the imam's impurity. However, one should be cautious about repeating this, as some scholars invalidate the follower's prayer if the imam's prayer is invalidated, without exception.