Hello, can you please tell me whether it is possible to play a BIAB style and change key in real time using chords played in the left hand of my keyboard? Thank you.

No. Band-in-a-Box reads a few bars ahead so the program can select what material to playback.

User Pipeline has created a program called "Live Arranger" that you might find useful. +++ HERE +++ is a link to a post where he talks about it.