Is it obligatory to repeat the Fajr prayer after doubting the correctness of the transitional "takbeer" (Allahu Akbar) following prostration, and later becoming convinced of its incorrect performance?

The place for the takbirat of transition is at the commencement of the transition. As for whoever differs and says the takbir after prostration, their prayer is valid according to the majority of scholars, because the takbirat, in their view, are a Sunnah (definition) that does not invalidate the prayer by being omitted or added. Therefore, it is not obligatory to pronounce the takbir of transition after prostration, nor is it necessary to repeat the prayer, whether this was due to forgetfulness or intention. This falls under the category of uttering a prescribed remembrance in prayer in a place other than its designated one.