The numbers from the Fluke sound about right. In simulation, I get 880 mVRMS across the capacitor, and 18 mVRMS across the 270 ? resistor. The simulated Zener is not a 1N4735, but another 6.2 V one, so the amount of ripple is only a guesstimate.
When trying to measure, is your Klein meter auto-ranging to the lowest (600 mV) range? If so, and you still read zero, then it seems to be defective.