Check ground connection from the battery to the engine block, from battery to chassis, and from chassis to cylinder head.
I think there is a ground that goes from the battery to the fuse box- but maybe not. That one might be the problem if it is. It seems to me that somehow one of the other relays- ignition or fuel is losing voltage and dropping out when you turn on your headlights. That doesn't jive with the battery loss of voltage though.
Have you used test mode to verify that your alternator voltage is remaining steady while driving? You might have one of those loose alternator plug-in connectors, or lost a wire in that harness. That would cause a good alternator to not put out.
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