Does your car have the LED tail light kit? I have a much older car than yours but I recently retrofit the caterham LED taillight kit and ended up with a very similar problem. I came to realize it takes very little current to get the rear blinkers to turn on. My car was previously wired to share a single dash indicator light for the turn signal, which was acting as a “short” between the two light circuits because the current was so low.
I would be curious if disconnecting the wires that lead to the speedometer turn signal indicator solve your problem? And if that does fix it, adding a couple diodes would allow you to reconnect it.