I know some of the old xr's do the same thing. I think this is quite common and may just be the way the bikes are. Adding a capacitor should help to fix the problem if it doesn't have one already.
What's your setup? Stock lighting coil? Aftermarket setup?
At idle rpm there's just not much juice produced. Flasher for the blinkers likely isn't seeing a high enough voltage to function. This is where a small battery setup comes in to keep things working when charging system output is too low. Many lighting coils area spec'd to give their measured output till something like 3000 or even 6000 rpm.
Below is for alternators, but generally a bike lighting coil setup would follow similar output curves.
With a 85 watt Moose coil my old KDX would not operate the flasher at idle (low beam and tail light on) until I added a small battery pack. Swapping to a LED tail bulb helped reduce power consumption enough to not need the battery, although the headlight was still noticeably dimming with each blink.
Is this a street legal Euro 'SR' model, or a typical KDX that has been dual sported?
Either way, it sounds like your system is not making enough power for the demands of your bulbs.
The LED bulbs in the taillight (and blinkers, too if you can find them) is a great idea.
You may also consider putting in a lower wattage headlight, or an off switch on your headlight so you can turn it off in daylight (may not be legal in your area).