My testing indicates that a 10ms delay was always too short and a 20ms delay was too short for close objects (which is counter-intuitive to me) - at least for my SR04 sensor.
Increasing the delays in my larger code base reduced but did not eliminate my false negatives. Testing with my actual head while as stationary as I could make it produced lots of false negatives, depending on what it bounced off. In some cases, there were no false negatives. Moving my head while not walking was problematic, too. I suspect I will get more false readings while walking because this adds sensor vibration.
To sum up: Minimum delay between pings for my hardware was 40ms. I have some inherent false negatives due to the target shape, target motion, and sensor vibration in my application. I'm going to keep using my digital filter, unless I've missed some other solution. I'd be happy to share my filter with others if it's of general use.