In case anybody is going to try something with the NTAG I2C Plus chip, since I designed the card I’ve had occasion to do further testing to compare the VOUT value at different current draws. Here’s a chart from the datasheet itself.
And here’s the data from my testing with various resistors on the NTI2C+ test card powered from an ACR-122U. As you can see the voltage droops considerably even at low current draw.