Solved.
It was not poor grounding or LED lights.
I finally had a chance to tow the trailer to a dealer (2 hours away). There, we were able to connect and tow an identical trailer with the Expedition. There were no trailer disconnects. This finally and definitively pointed to the trailer as the problem.
To make a long troubleshooting session short, we found that the resistance of the trailer brakes, as seen by the truck, on the "good" trailer was about 1.7 ohms. On the "bad" trailer it was about 2.0 ohms. That does not seem like it would make much difference, but as an experiment we decided to lower this resistance by adding a parallel resistor (across the brake control line to ground).
I pulled from my EE degree 40 years ago to calculate the needed resistor should be 0.8 ohms or so. Of course we did not have a resistor that size, so we used a 10 W dashboard lightbulb.
We ran a wire from the brake line all the way into the truck cab. Then, took a couple of different light bulbs with us. While I drove, the mechanic would connect the light bulb using the ground from the 12V plug-in power port on the dashboard.
Sure enough, when we used the 10 W bulb, the trailer stayed disconnected. In fact, he could force it to "disconnect" by removing the bulb from he circuit.
Problem diagnosed.
The solution we implemented may be temporary, but it worked the entire 90 mile drive home! We simply wired a bulb socket between the brake control wire and ground underneath the trailer. Now, whenever I put on my brakes, a small light bulb is coming on under the trailer ;-) The good point is that I can pull that socket and wire in a small resistor to make a more elegant and permanent fix.