Compensating for offset voltage in bridge measurement before instrumentation amplifier
I would try to use indirect current-feedback instrumentation amplifier. Like MAX4208/4209 or AD2420 or any other....instead of using the classic three opamp instrumentation amplifier. These amplifiers are made to cope with the issue you have. 

Link:


  The traditional 3-op-amp in-amp architecture, shown in Figure 2, has a
  differential gain stage followed by a subtractor that removes the
  common-mode voltage. The gain is applied on the first stage, so the
  offset is amplified by the same factor as the signal of interest.
  Thus, the only way to remove it is to apply the opposite voltage to
  the reference (REF) terminal. The main limitation of this method is
  that adjusting the voltage on REF cannot correct the offset if the
  first stage of the amplifier is already saturated
  
  
  
  Indirect current feedback (ICF) in-amps, such as the AD8237 and
  AD8420, make it possible to remove the offset before it is amplified.
  Figure 3 shows a schematic of the ICF topology