Hi,
Most likely you have moved on but you might consider using a "instrumentation" amplifier IC in place of the three amplifier circuit. The instrumentation amplifier has closely matched internal resistors for better performance over time and temperatures. You would then set the gain using only one precision resistor. That resistor should be a 0.1% with 25ppm/C as a min. Those cost are still affordable.
Also, if interested, I'm working on a higher precision analog to digital converter (DAC) for Arduino. It's a small shield with a 22bit DAC. The shield has a trimmable voltage reference which would allow you to connect your strain circuit and trim the DAC output to a Zero value. The DAC also has a built in trap filter for 60 Hz noise found in most labs.
Cheers