What is required of me after having performed 'Umrah on my own behalf and Hajj on behalf of my deceased aunt with the intention of Tamattu', then forgetting to make the intention of circumambulating on behalf of my aunt during the Farewell Tawaf?

Since you performed 'Umrah (definition) in Shawwal and Hajj in the same year, you are a mutamatti‘ (one performing tamattu‘ Hajj). It does not matter that you performed 'Umrah on your own behalf and Hajj on behalf of someone else, because tamattu‘ does not require that both rituals be performed by or for the same person. In this case, you are required to offer the hady (sacrificial animal) as is incumbent upon a mutamatti‘, or fast if you are unable to do so. Your farewell tawaf (definition) is valid, even if you did not explicitly intend it on behalf of your aunt, because the initial intention encompasses all the rituals of Hajj.