All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, is His slave and Messenger.
According to the majority of scholars, transitional Takbeer is recommended. Based on this opinion, there is no problem with what you did and you do not need to perform Sujood As-Sahw (prostration of forgetfulness) in this case. According to the Hanbali scholars, transitional Takbeer is obligatory and should be done during transition. However, some Hanbali scholars, like Al-Mardaawi in his book Al-Insaaf, held that the preponderant opinion is that there is nothing wrong with starting the transitional Takbeer before the transition or completing it after the transition because there is much hardship in avoiding doing that. You can follow this scholarly opinion because it is easier. Hence, you should keenly recite the transitional Takbeer in its due time (during the transition). If you fail to do so, it is pardoned and you do not need to perform Sujood As-Sahw.
Allah Knows best.