The ignition system is totally separate from the charging system so the battery won’t have an effect. Have you tried checking the stator windings for starters? The readings are in the Haynes manual. Also although I don’t think it is swap out plugs and caps. CDI is not so easy to test but I have a couple you could try. I went through similar symptoms to yourself including a rewind of the stator and never got to the bottom of it. Eventually fitted a Power Dynamo and now half a kick and runs sweet as.
my ten bob is on the stator windings breaking down, there has been a few posts mentioning this before, good when cold but after a few miles they deteriorate, can be tested when cold and then after warming up with a hair drier they have a different value.. The CDI? they either work or not in my experience, but either way, they are now 40 years old...
I'd drop mutts a message, i got mine from him, too.
Measuring the resistance of the stator when hot will always be different to when ambient temperature. This does not mean it is broken! If it goes open circuit or dead short when hot, then it is fooked! :-)
"Every day spent above ground is good. Every day spent on two wheels is better."