Which opamp to use for differential reading of a shunt resistor?
breadboard-compatible opamp which is great for this purpose?


Low-noise measurements and breadboard – might be a bit hard. Breadboard is generally know for the following properties:


bad connection reliability
high line reactance, making it hard to observe fast-moving signals at all
high cross-talk between neighboring lines
relatively high resistivity at contact points.


In other words, when dealing with mΩ resistors, you simply can't use breadboard, the unwanted effects can very easily overshadow that.

Then, you're really driving your Opamp rail-to rail – that might make accurate measurement very hard. 

I'd use a boost or inverting SMPS circuit to simply increase the supply voltage of my opamp. That way, I could use a single shunt for both directions, since I could guarantee that the top and bottom voltages of said shunt would be well within the input range of the Opamp.