I think the specifics of the magnet winding must be a key element to making this work.
A relay winding has about 150 ohm resistance and so will draw about 80mAmps from a 12 volt battery. But A23 12volt batteries only provide about 15mAmps peak. The 3volt battery in the BadAss wouldn't produce anything like enough Field Strength in a standard relay winding, so something else is going on. Either that or the Giant Entour (Yamaha) uses a much higher field strength than other bikes.