I'm no expert on lavenders but I think lavender x splash would give you the same as black x splash, all with the lavender factor but none showing it. If you back crossed those birds back to a lavender you'd get 50% lavender and the rest blacks, blues and splashes.
Meaning the splash x lav chicks would only have 1 gene for lavender so it wouldn't show but if you back crossed to lavender you'd get 50% lavender chicks and 50% with just one lavender gene that would again not show.
Splash rooster to lavender hen = all blue chicks that carry a lavender gene.
Cross a cockerel back to lavender hen you could get black, blue, lavender and blue/lavender (don't know what that last one would look like.) some black and blue could carry one lavender gene.
Bred chicks together and get the same thing plus splash, splash carrying lavender and splash/lavender.