Just for reference sake - here are pictures and measurements of the fix. It took about 30 minutes and cost less than one dollar.
Mercedes recommends installing a 220 Ohm, 1.5 Watt or better resistor in parallel with each of the turn / brake light circuits.
First some physics - when you install resistors in parallel with an existing load, you lower the resistance of the overall circuit. The formula is 1/Rtotal = 1/R1 + 1/R2.
I measured the resistance of each brake light circuit from the Airstream's trailer connector using a fluke digital multi-meter.
Both left and right turn / stop circuits measured 2.2 Ohms with incandescent bulbs installed, and 29.8 ohms with LED bulbs in place. This implies each of the two LED bulbs on each side has roughly 60 Ohms of resistance, vs. roughly 4 ohms for an incandescent bulb. For comparison, the backup lamp circuit measured 4.2 Ohms, and the running light circuit 1.8 Ohms. The estimates for the bulbs above are all rough - I was too lazy to measure bulb resistance directly; all of the measurements above also reflect the resistance of the trailer wiring.
So using the formula for calculating parallel resistance, when installing a 220 Ohm resistor in parallel with a 29.8 Ohm existing load, 1 / Rtotal = 1/220 + 1/29.8 = 26.3 Ohms. So we expect each turn / stop circuit to measure about 26.3 Ohms when the repair is completed.
The first picture attached shows a 220 Ohm, 5 Watt, 5% tolerance, metal wound resistor. A pack of two of these cost 89 cents from our local Fry's Electronics.
The second picture shows the resistor in the middle of being installed. For ease, I chose to install inside the tail light fixture.
First remove the outer tailight lens. Then gently pull out the reflector. You'll see three wires - a white ground wire attached to the back of the reflector, and two wires (in my case yellow and brown) attached to the bulb socket.
The only trick is that you have to determine which of the two wires feeding the bulb is the turn / stop signal. I did this by connecting a wire to my battery and then to the appropriate pin on the trailer plug. It turned out the yellow wires are for the turn / stop signal.
As you can see in the picture - I simply removed a bit of insulation from the yellow wire with a hobby knife. Then I connected the resistor between the exposed copper wire and the white ground on the back of the reflector cup. The picture is before soldering and covering the connections with liquid electrical tape.
After screwing everything back together, each circuit now measures 26.8 Ohms - which is close enough to the expected value that it shows the resistors are correctly installed.
Final test is to apply 12V
to each of the brake / stop circuits - and both sides illuminate correctly.
Later this week I will have a chance to hook up at night again and verify the fix actually stops the GL from flashing all the lights.
So what is actually going on here? I have no idea. The resistance numbers I measured above are for the incandescent bulbs when cold. When they are illuminated and get hot their resistance will increase - but only to 7 or 8 ohms per bulb. So the 220W resistor is not trying to make the whole circuit look like it has the resistance profile of incandescent circuit. Anyone know what the tow vehicle is actually trying to measure ?