After having tested the bra and worn it for a day I noticed tension pains in the left trapezius muscle (from the neck down towards the shoulder), and I think the weight of the bra is the culprit. I have therefore decided to bone the cups and see if this helps. I have looked at boned bras from the era, and they have bones at the sides of the cups, as well as two bones in a reversed v placed in the cup, like this one from Etsy:
Update November 14th:
I ended up using artificial whalebone (sturdy plastic), and it worked very well. I lined the bottom of the bra front with a velvet ribbon to protect the skin from poking. This is rather important as the bone ends will dig in pretty bed without the velvet ribbon.