I do suspect your rooster is Recessive White, since no white chicks were produced. So, you could produce more white chicks through breeding them together for a short period. After that, new blood would have to be introduced.
Your white rooster is most likely carrying barring, at least one copy, resulting in the one barred chick.
Lavender is recessive, so first generation chicks won't show it. They'll be split for the gene, & would have to be bred back to the hen to produce more lavenders.