Final update. I have a 6/12 volt battery charger. Set at 6 volts, it puts out 8 volts according to my multimeter. Hooking the LED tail light wire up using the charger as the power source but still mounted on my truck and using the new ground wire I ran, I get a much brighter light. Set at 12 volts, I get a really bright light.
Hooking up the LED brake light wire with the charger set to 6 volts (8 volt output), I get a light that blinds me and leaves spots in my eyes.

My conclusion is that these LED lights need more voltage than my 6 volt battery can generate.

With my engine running at a fast idle, the voltage at the tail light is 7 volts. The tail light is definitely brighter than with the engine off ( I can actually tell it's on from a distance in the daylight). It may be good enough. The brake light is pretty darn bright, so I'm going to wire them in and see how it goes.

Thank everyone so much for your patience and advice. I feel like I've taken a course in electrical troubleshooting 101.

My dad always told me, "Don't force it, get a bigger hammer".