Cant help you on champagne screw off caps. I do know that normal American champagne or sparkling cider bottles can take a standard bottle cap. I have no experience with the screw on ones.
Martinelli's Sparkling Cider bottles work great. As do most American sparkling wines that come with the standard mushroom cork or the plastic version. Those will take a 26mm standard bottle cap if your capper will fit the bottle. Mine will do that just fine. I have the Red Baron Beer Bottle Capper.
Primary: 3 gallon cider S-04
Secondary: Valentine Apfelwein Clone 1 Gal
Bottled: Halloween Apfelwein Clone 1 Gal
On Deck: 90 Shilling Clone