Yes a bad battery will do at a push as it will soak up the overcharge from the mag. But you will still be working the regulator hard and from personal experience they are rarer than rocking horse sh1t and twice the price.
Right had a good look at the wiring and the ignition system is completely self contained on your bike, the system runs very similar to points.
The carging/lighting circuit has no connection to the ignition system whatsoever, it generates its own power, but the voltage regulator will struggle without the battery to soak the excess in the lighting/charging circuit. And i repeat they are expensive and rare, it took me 6months to find one.