Stay the course with the White cock. Changing cocks will only confuse the issue as the sperm from the White can stay in the hen for two weeks or more. You have a better chance to make Pied WE with the White cock. Using the BS Pied WE cock will also produce more Dark Pied, split to either White...
Dark Pied is a term given to a bird that carries two Pied genes. It can have little to no white showing visually, adding a White gene will make the offspring full Pied showing some to a lot of white on the birds. Silver Pied is made up of one White gene, one Pied gene, and two White Eye genes...
The chick is Pied, whether it is WE remains to be seen. I have no idea what color it is as I don't know the color of the parents. The father is White pattern, he contributed one White gene and the mother has to carry Pied, (one Pied and one White gene or two Pied genes) and donated one Pied...
White and White Eye are two different genes, with either it takes two of those genes to produce full pattern, if only one gene they are considered split to that pattern.
What you are calling white dots should actually be a white frosting on the tips of the feathers. They are usually on the top...
Telling by the white throat patch, white primaries, and overall silvering it is defiantly carrying WE, perhaps a full WE. Seeing that it is from a Pied pairing it is likely split to Pied or Silver Pied. Mating her to a White will give you indicators of what she is carrying.