Resistance measurement using high side current sense amplifier
Yes, this will probably work. High side current sense amplifiers are just differential voltage amplifiers, usually with fixed gain, or gain that is select-able over several discrete options. After glancing at the datasheet, it seems this one has three gain settings.

The current sense amplifier does not "force" current through anything. It just measures the voltage drop on the shunt resistor, rsense. Where the datasheet shows a load, you would connect your precision current sink.

This part does have an output, and in the datasheet, they show how to use the output to create an inrush current limiter. The set pin and voltage divider would be utilized for this.

The basic technique for low Ohm measurement is to force a known current through the part in question, and measure the voltage developed. I usually just use a lab supply and a volt meter. If you want to be really accurate, you can measure the current as well as voltage (if you have two meters). But I usually just assume the lab supply display is accurate enough.

Final note is that the offset voltage specification of the current sense amp is not that great (considering the small full-scale input voltage). I recommend that you pulse your current sink on and off. This will cause the current to turn on and off. On the microprocessor side, you will get one voltage reading from the current sense amplifier when the current sink is on. Call that Vhigh. Then you will get another reading when the current sink is off. Call that Vlow. The difference is the true voltage you are trying to read. This will basically calibrate out the offset error.