I'm still not able to well understand what you got. But anyway, you say you have a structure AABA. Are you in unfold mode, that is have you only one big 'chorus' of 32 bars? In the main page of BIAB, bar beginning of chorus should be set to 1, bar for end of chorus should be set to 32, and number of choruses set to 1. Did you copy your second A section from the first A section ? As I previouly said, in Satin Doll, you have temporary modulations, but globally the key signature of the song will stay in C. The display of the key signature don't reflect the actual key signature of some bars of the song. So, when BIAB plays /E-7 A7/ the display of the key signature will not move to D.
As far I understand, when you play the second A section - with the same chords as the first section, you just get a C chord for the 8 bars, is that correct ?

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