Barring and cuckoo can be considered the same for genetics purposes--the difference seems to be in how tidy the white barring is. But they both use the same barring gene on the same black background.
For barring (or cuckoo), it's probably more convenient if you use a barred rooster, who can sire barred chicks of both genders. (Although using a barred hen could produce barred sons, not pure for the barring gene, who could in turn produce both sons and daughters with barring and without barring.)
For the single lacing and double lacing, I don't think it matters what gender bird you bring in. So you could use a purchased rooster with one of your hens, or a purchased hen with your rooster.
And of course if you purchased one hen in each color to add to your current flock, you would only need one breeding pen that year. (Although then you would split them up the next year, to cross various colored hens to their own sons.)