The LED idiot light
might not work correctly with LED turn signals.
As I understand it, current flows through the idiot light in both directions, depending on which side is blinking; the other side serves as a ground path for the idiot light, with the current apparently being low enough to not illuminate the lights. This bi-directional polarity is why a polarized LED does not work properly in this particular idiot light.
If you have LEDs in the lights and the idiot light current is the same polarity as the lighting circuit on the non-flashing side (which it seems to be from the "stock" diagram below), the idiot light
should work but the LED lights on the "off" side
might blink too. This is what happened when you had the regular light bulb in the idiot light.
The solution would be to rewire the idiot light with two diodes and its own ground path, so that it is unaffected by, and will not affect the "off" side of the turn signals. The other advantage of doing it this way is that you can use a polarized LED, which is what all of the smaller ones that fit the socket are.
Here is a diagram of the stock turn signal circuit:
Here is a diagram of the circuit rewired with the diodes:
Disclaimer: I have not done this, but I have researched it and have some basic electrical knowledge. I'm not claiming that this is how it will work, this is what I
think will happen.