Can't Post pics right now.
A beer bottle cap (you use the term "regular bottle cap") on top of a euro champagne bottle won't work. It will be too small. The Euro Champagne Bottle needs the larger 29mm cap.
You don't even need a new capper either, just a new "Bell" for the capper.
For handheld models, the bell threads onto the magnet post thing. No tool, just hand thread the old one off, and hand thread the new one in.
Then take a flat head screw driver and push out the metal plates that grip the neck of the bottle, and re-insert "Backwards." This will accomodate the larger neck of a 29mm bottle. I think on my red baron there is actually a 26 stamped into one end of each of these plates, and a 29 on the other end.