I think @insightful is on to the trouble. The ground for that side has a connection problem so there is added resistance to ground. When the higher current for the brake light is needed then there is more voltage drop across the added resistance of the bad connection to ground. That is really noticed in the running light operation which has a higher resistance compared to the brake light. To prove there is bad ground you could run a jumper lead to the tail light ground and to a known good ground point. If the lights then work normally then you have proved the cause of the problem.