Then to get better type I could breed the mottled lavender to black and get all black chicks split for mottled lavender, and breed those back to mottled lavender to get half mottled lavender and half black splits? Or would the ratio be different?
That's a good plan for improving on type(if the blacks have the best type). But remember lavender and mottle are separate genes.
Breeding the black split for lavender AND mottle back to a mottled lavender would result in:
Half of the chicks will be lavender, half will be black split for lavender.
Half of the chicks will be mottled, half will be split for mottle.
which means you will be getting mottled lavender, lavender(split for mottle), black(split for lavender and mottle) and mottled blacks(split for lavender) in about equal percentages.(1/4)