This afternoon while I was checking things out myself, testing fuses, checking grounds, checking for wires disconnected from the back of the fuseblock, etc. it still wasn't working. Then I started rearranging wiring to work better with Tom's battery bracket, unplugging, moving and replugging things. When my buddy showed up the other circuits were just working. Maybe I didn't have a connector together all the way. But the headlight, brake light and horn work and the starter turns over. As soon as we turned on the turn signals, the 10A fuse for that circuit would blow (sometimes we'd see a flash of turn signal light). This behavior was completely reproducible and as long as we didn't touch the turn signals everything else works. He tested for consistent ground and power throughout the system and it all seemed to check out fine.
So I'll get a relay and hopefully that will be the end of it. I'll probably try that headlight module out again in a week or two.