Once B1 gained possession, then A's team control ended. The tip by A2 does not reestablish team control, so there is no back court violation. Once A3 touches the ball (if she is standing in her backcourt) the ball has backcourt status, thus the ball cannot still be in the frontcourt. She can begin her dribble and she does not fully have front court status until both feet and the ball are across the division line.
Maybe the key discrepancy here is whether B1 actually gained possession, or if it was really just a loose ball so that A's team control never ended. If that's what happened, then it would be BC.
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