Do current / power monitoring ICs (e.g. INA219) allow frequent re-reading?
First, your estimation of the speed of the I2C bus is off. 1uS is one cycle at 1MHz, and the I2C bus in high speed mode is only 2.56MHz maximum. So in 1uS you will be able to transfer under 3 bits over the I2C bus. In 4uS you will only get about 10 bits over the bus, so not enough to read a 12 bit value + transaction overhead.

I have worked with this part before, and you can read it that fast. Use short traces and don't add too much (or any) capacitance between Vin+ and Vin-.

The 0X11 and 1000 settings use the same ADC, the "ADCx" in the table are bit names, not different converters. The two settings are the same, because it appears that clearing ADC4 reduces the resolution by a number of bits by 3-[ADC2...ADC1] while setting it increases the number of averaged samples by 2^[ADC3...ADC1].