Rear signals won't flash

[JMD]

Hi all,
Almost finished with my '68 k10 frame off restoration except for an issue I'm having with the rear turn signals. All the wiring is brand new, all the lights are LEDs, and I have a no load flasher for both the turn signal and hazard circuits.

The issue is that the rear turn signals won't flash if the running lights/headlights are on. The front signals work fine, as do the indicators on the dash. If I have the running lights off the rear signals work just fine, but as soon as I pull the headlight switch out the brake light stays solid.

Before I start tracing wires and threatening to light the thing on fire, is there anything I should check right off the bat?