I assembled two of the test boards, one with ST25DV and the other with NT3H2211, then took some measurements with different loads.  

Turns out the ST25DV does not regulate the output voltage, and it drops off with load fairly steeply.  The NT3H2211 is better regulated, and maintains the voltage better when loaded.

The CH32V003 needs at least 2.7V to run and will consume between 2.3 and 1.9 mA depending on how many peripherals are enabled.

The measurements are listed below and a plot of the output voltage vs load curve fit to the measurements.  This shows that the NT3H2211 provides a much more consistent voltage and stays above the minimum 2.7V up to a higher load current.  This suggests it will be easier to power the CH32V003 and a few peripherals with the NT3H2211, but there is not much margin.  

I hear they are coming out with a low power CH32L103.  I wonder how the price of that will compare to the CH32V003.  It is available in the same TSSOP20 pin package, but it has more memory and advanced features like USB, which is not a requirement in this case.