Finding ground in a battery powered circuit?
If you have +5V and -5V then half way between supply voltages is 0V i.e. the common connection of the two batteries - this should be regarded as your "ground" (or 0V) for your circuit. If you read what TI say about the TLE2426 you'll possibly agree: -


  In signal-conditioning applications utilizing a single power source, a
  reference voltage equal to one-half the supply voltage is required for
  termination of all analog signal grounds. Texas Instruments presents a
  precision virtual ground whose output voltage is always equal to
  one-half the input voltage, the TLE2426 “rail splitter.”


You have two power sources namely +5V and -5V and these are likely to have a third wire which you have available that does exactly what the TI chip sets out to do.

Why did the INA129 become unbalanced: -



Using a resistor divider to feed the REF input means that you are altering the 40k resistance at pin 5 and this will unbalance the ratios of resistors and lead to common-mode signal problems.