I am not 100% sure how the genetics of colour work in araucanas. However, the old way to get sex linked chicks was to cross a white hen with a coloured cockerel. Then the cockerels were white, and the female chicks coloured. This was traditionally done with Light Sussex (ie white) hens and a Rhode Island Red (ie coloured) cockerel. So you then knew on hatching that the brown chicks were pullets and didn't have to raise a lot of white cockerels that you didn't want.
So assuming the system works with araucanas, then your white chicks will be cockerels from your white hens, the lavender chicks will be a mixture of pullets from your white hens, and both sexes from your lavender hens, and your reddish brown chick (most likely) from your black hen.