Hi Henry,
Which revision is this? (it's on the SMD side, lower left).
Do you still have the calibration values for it? That might help in case they need to be changed.
It might be possible the circuit wakes up too fast, causing a brownout during auto-ranging.
What happens if you add a 10uF capacitor to your power rail in your circuit? How about a 100uF? (this to the regulated power supply that powers your ESP/attiny).
If you take a simple 10K variable resistor (trim pot) and add a 50-100 ohm resistor in series with it, and slowly ramp up from uA to mA range, there should be no flicker, the CR should range smoothly. That's how you know the CR is functional and does its job. As far as how fast it can do that, that is a separate issue and it can be made to switch faster at the expense of some extra noise (seen more evidently on the OLED if you have one, but not on a RMS DMM). That involves sampling the ADC faster.