I am wondering if your problem it isn't the input voltage range. If your sensor circuit is float, then you could have problem with amplifier. I assume that the voltage you have written it was differential voltage. But what are the absolute voltages on amplifier inputs?
With INA129 and your supply you should have not more than +-10V on any amplifier input (internal too). Instrumentation amplifiers are very difficult to use becouse of this. Before I use any amplifier first I simulate it, so maybe you should measure absolute voltages on both inputs, then simulate it (I use p*r*o*t*e*l) using discrete amplifier and circuit from datasheet. For simulation purposes I always use much higher supply voltages, and then I could check every point.
If problem is with floating sensor you could try to ground it, and sometimes large resistor is enought
regards
KamW