Ok,
I am pleased to report that changing to a different brand and model of servos has eliminated the chatter. Some servo models (Hextronik htx900) must be more sensitive to slight deviations in the signal than others (Tower pro MG996R).
I'm puzzled why the distance reading by the SR04 sensor would have such a profound effect on servo chatter, but for the first type of servo I tried, it does.