I don’t quite understand your problem. If you broadcast message as an input to your operator that depends on this configuration, each instance of your operator will receive this configuration. It shouldn't matter whether Flink scheduled your operator on one, some or all of the TaskManagers. It only should matter if operators running your configuration sensitive code receive the broadcasted message.
DataStream<> input = xxx;
DataStream<> controlConfigInput = yyy;
DataStream<> data = "">
Or slide 36 from here: https://www.slideshare.net/dataArtisans/apache-flink-datastream-api-basics