little more troubleshooting...
In my earlier test with a separate battery, I left the antenna attached to the tailgate mount simply because I knew it wouldn't be a valid test if the antenna didn't have a groundplane to work with. I thought maybe that was a bad test since the radio's ground wouldn't be the same as the antenna's ground.
So today I decided to repeat that test, but with everything completely disconnected from the vehicle, including the antenna. To get around the groundplane problem I fashioned a makeshift dipole from my 2 firestik antennas. (One connected normally, with the other pointing in the opposite direction with the base in contact with the coax connector barrel.) I just held it like this in my hand at the feedpoint. When I brought it close to the engine, I did pick up the same noise. As I stepped back, the noise went away. I was also able to pick up some noise near the rear of the vehicle, but this was hit or miss. This tells me that my problem (or at least some of my problem) is 100%, without a doubt, radiated RFI. It could be caused by current flow on parts that are not well bonded though.
I tried touching a wire from the alternator to the battery, and that didn't seem to help at all. (But I haven't tried doing that with a true ground-strap torqued down.)
I'm thinking maybe I will go visit a CB shop on Friday and see if they have any other ideas. Short of adding more ground straps, I'm fresh out.