I made the changes to the bodice and put it back together on the stand. It looked good!
I made bias-binding from the top fabric and finished off the edges. I made the bias binding fairly wide, approx. 2cm. Although this is wider than the 1930s dresses I researched, I did this on purpose so that if an edge gaped a little, or came away from the body, the white backing fabric would not be immediately visible. It may look less delicate but has the desired function.
What I did discover however, after putting it all together, was that one of the back bodice panels was wider than the other! Extremely frustrating, but I measured many times to check and decided that the difference was noticeable: I would have to un-pick and trim some of the excess away so that they matched. I did this, and upon scrutinising the bodice once more, they still seemed slightly out! In fact once I re-measured the difference was extremely slight; and due to the fabric stretching in places. The bias drape was constantly changing the nature of the fabric itself! Nothing to be done; luckily the style of the dress really exploits the draped feel, and I don't think that it's overall very noticeable.
More lessons to be learnt here regarding handling bias drape projects.