I have used a Li Ion battery to test dyno powered lights. At roughly 3.8 to 4.1 volts DC, their voltage was enough to light up the lights without any danger of damage. At that low voltage, they are not really bright, but bright enough to see.

For testing I used a phone battery from the era when phone companies allowed consumers to change their own batteries. I also have removable Li Ion camera batteries that I have used for testing.

If the lights work when testing this way, I would try checking the contacts again, perhaps cleaning off the contacts with rubbing alcohol.

If you have access to another dyno powered headlamp, you could swap that out to see if that solves it.

I think all Shimano dynohubs are grounded to the fork, I know that some are. Is there any possibility that some of your wiring has an intermittent ground to the fork or frame?

I can't think of anything else that SteelBikeGuy did not already address. He had a very good comprehensive list. Good luck figuring this out.