My bet is headlight breaker, part of the switch.
I'd start by pulling fuses to see which (if any) make the click stop.
It's a pretty big load to make the ammeter flinch.
Definitely disconnect the battery when not nearby.
Headlight switch. Make sure you have a ground strap on the column going to the studs that hold it in place, and a ground strap between the firewall and the cylinder head. I've installed aftermarket headlight switches that had their own stud explicitly for connecting a dedicated ground wire to them.