Using multiple shunt amplifiers with a single shunt resistor
My questions are:

Is it sane to put multiple amplifiers on a shunt like this?



Sure, it is perhaps a bit expensive compared to changing the gain of one amplifier, but I see nothing else wrong with it. 


  And if not, which alternate scheme should I be looking at instead?


Changing the gain of the instrumentation amplifier would likely be more cost-effective. There are some with selectable gain or you could switch external resistor networks. 


How much will this affect the readings and the load?



There should not be any noticeable effect on the reading from operating several amplifiers as shown provided you stay within the rails on all of the amplifiers. The source is extremely low input impedance, so effects from bias current, input noise current, and offset current will be way below the noise floor.