Radio Interference

This is one of the reasons I never took the plunge for LED for my old iron, especially ones with working radios. ............I've discovered the importance of FCC compliance. While FCC doesn't apply here, again I've discovered LED bulbs "can" cause EMI in radio reception..............

.....................Somewhat mis-worded? Maybe you meant this:

First the LEDs themselves don't cause RFI, unless involved in transmitter antenna field, which is a whole different thing

Second the FCC IS involved, or "should be." But not in automobile aftermarket LEDs. The thing that is "FCC" so to speak is (home) AC powered LEDs which use a SWITCHING POWER SUPPLY and THAT is what makes all the noise, THAT is what is supposed to be certified (and often is not) None of that applies unless you are talking about the high output HID headlights, which DO use a switching supply.

So far as the OP's problem, just not sure.

Might be defective antenna / coax, or not properly plugged in?

Check radio supply voltage.........Maybe a wiring mistake is "dropping" voltage down during the flasher operation.

Otherwise you can try some bypass caps across the flasher. "The problem" in car wiring is packaging them so they are both robust and yet "short wiring leads" to the flasher.

Sometimes simply relocating the offender helps. Keep the thing "dressed" close to a ground plane.........the metal dash. Try separating the radio power wire from the flasher wiring.

You can also install additional filtering "into" the back of the radio, although it normally should not be necessary.

If you can't do anything else, go to the parts store and get them to look up in the "picture book" (numerical / pictorial listings) a generic radio supression cap. NAPA shows them. Install that as close to the flasher as you can get