There is a sex-linked gene that allows dark skin, and is recessive. So the rooster could be carrying that gene, and it would be visible in about half of his daughters. Why he would be carrying that gene is a good question-- maybe a cross with Silkie in the distant past? The dominant traits from Silkies can be bred out in two generations or so, but recessive ones are harder to get rid of.
Alternately, the chicks could have a different father and/or mother than expected. I think different father is more likely, given the sex-linked inheritance of that gene. But chicks from another hen could easily have another father as well.
If anyone is breeding Paint Cochins, there are going to be some Dominant White Cochins that exist as well.
But it's easy enough to prove as regards a specific chicken: breed the white chicken to a solid black chicken that does not carry recessive white. If you get a bunch of white or paint chicks, you have a Dominant White.
Alternately, there is now a blood test to see if a chicken has recessive white.
https://iqbirdtesting.com/whitegene
I have no personal experience with this test, but assuming it is accurate, it would be faster to test one bird that way rather than test-mating the bird and hatching eggs to see the chicks. But the test might be more expensive than doing a test mating.
It looks like they don't have the price on that page, but it is listed on the main page:
https://iqbirdtesting.com/
It's showing $50 or $55 for each bird tested, with different kinds of samples costing different amounts (blood sample is cheaper, feather or eggshell sample is more expensive. The eggshell is supposed to be the one the bird hatched from, not one laid by an adult hen.)