You can get white purebred, registered Soay from a few breeders, although more usually they will have some spots and coloured marks. White is a natural colour for Soay but they tend to get picked off by the bonksies on St Kilda. My first spotty white lambs appeared from black ewes bred to a black tup, and each had a black twin.
I think using your white crossbred tup would give you 50:50 white:mouflon lambs - the tup will have recessive mouflon genes hidden by the texel white, so half the lambs would inherit the mouflon gene and half the white, in the first cross.