Rover,
With the additional data I can tell you exactly your problem.
Using the devices you selected your results are within their specifications.
Particularly the MAX4173H, input offset voltage is 3mv max to 5mv over temperature.
Add the gain back 3mv becomes 300mv, swamping out your zero reading.
Typical offset error is 0.3mv,
Shunt current of 300ma reads 0.3mv, same as the offset error.
Yep, this system is incapable of reading 300ma in shunts.
The error signals must be below the signal of interest.
Keep input offset voltage less than 30uV, to read 300vV signal.
The MAX4173H integrates several components, a discrete version can be built with more precise parts. Its gain accuracy is 0.5% typ, 2.5% max at 25C and 4% over temperature.
Let me know if you need help to do so. (or intend to)
Have fun,
Scott Beversdorf.