Is my instrumentation amplifier's voltage offset causing problems?
Why don't you get an ac signal source, attenuate it 500:1 with a 499 ohm resistor and a 1 ohm resistor and insert this into one grounded element of the bridge. DC offset introduced can be countered with another 1 ohm resistor in the bottom leg of the other grounded element.



Also shown is a method that doesn't break-into the bridge - it applies a high value resistor across one of the grounded bridge elements in order to simulate a shift in bridge output. This can be controlled with a fet fed with a squarewave as shown. It's even possible to use an opto-coupler in this type of application should the return line of the bridge be very sensitive.

Now you can inject (say) 1kHz at a known measurable amplitude and see what comes out from the instrumentation amplifier. If the AC gain is fine then you have to look at the dc issues that may be causing your problem BUT until you rid yourself of the doubts about ac gain you might keep going round in circles.