Post up some full pics, I love my Lambretta.
Having never worked with the BGM CDI, I guess the first question is: is this brand new? If so, I would suspect that the coil for lighting is bad. An easy way to test this is check resistance between the yellow lead and ground. Really high resistance or lack of cont. would indicate bad coil. If it's new, I would contact manu/retailer to get info on replacement.
I would also check the ground on the regulator/rectifier to make sure it's just not poor contact with stator ground, as your paint looks very new/nice, it could just be a painted over contact.
Barring any of this, I guess the first question is, have you test voltage while at operating rpm? You should see a fluctuation in output with RPM at the yellow wire direct out of the stator. If not, I would highly suspect stator being bad. If if fluctuates up to a decent nominal voltage there, but not after reg, than reg may have issues, i.e. grounding or failed.