Yar.
Sidelight 5 Watts at 12Volts.
LED probably a tenth of a Watt or something like that so ZFE thinks the bulb has gone (which it has). Don't know where the ZFE threshold lies but you probly need to burn something like say 3.5 Watts.
Using the formulas V=IR, P=VI we have R=V^2/P so R = 12x12/3.5 = 41 Ohms.
So you need something like a 40 Ohm load that can dissipate 5 Watts to connect in parallel with the LED.
Suggestion: Get a pack of three 100 Ohm resistors rated at 2 Watts each. Connect one in parallel with the LED. See if this fixes it. If not, add a second in parallel with the first. If this doesn't fix it add the third.