I think that this is what I have come across before, where the light assemblies share the same LEDs for brake and running lights, and just run them with more current for braking. The issue is that there is a single electronics module, and that runs from whichever of the input is at the highest voltage. The module senses the voltage of the brake light wire, and if that is high enough, the module runs the lights at high brightness. Light assemblies that can run on 12 V or 24 V are more likely to be made like this.
Here is an example of a light fitting that does that:-
https://www.ringautomotive.com/en/product/RCV4800 It uses the same LEDs for brake and tail lights. The other lights are separate.
The problem comes when the brake light voltage is high, but slightly less than the running light voltage, then just about no current will be taken from the brake light supply. Then your indicator flasher relay will not sense any current to the rear lights and will flash fast to indicate a blown bulb.
On those lights, this is what happens:-
Running light wire 12 V, brake light wire 0 V. Light dim, low current taken from running light supply.
Running light wire 0 V, brake light wire 12 V. Light bright, larger current taken from brake light supply.
Running light wire 11 V, brake light wire 12 V. Light bright, larger current taken from brake light supply, no current taken from running light wire
Running light wire 12 V, brake light wire 11 V. Light bright, larger current taken from running light supply, very little current taken from brake light wire.
If the wires are both the same voltage, the current could come from either wire.
On your motorbike, I think that the flasher unit may be causing a small voltage drop, so that the brake light wire on the lights running as indicators are running from a slightly lower voltage, and all the current come from the running light supply.
If I'm right, you can put a diode in series with the running light feed to the rear lights, in the direction that keeps them working. Then the voltage to the running lights will be a bit less, and the larger current will come from the indicator supply, as it does now when the running lights are off, and the flasher unit will detect the current and work normally.