Paint is just one copy of dominant white over black? Just take a bird that is dominant white and mate them to any black bird and you'll get all paints. Lavender isn't part of any recipe. You can include lavender for a lavender paint (resulting bird will have grey instead of black spots) but that's not too common.Color has really nothing to do with if they are pure silkies. And paint plus paint doesn't mean you'll get all paints.
paint has a genetic 'recipe' to accomplish, I think lavender is usually part of the recipe.
Maybe this can help.
https://www.pto.ca/forum/viewtopic.php?t=2793