Art,
The issue is that by connecting pin 15 to pin 4 you attempt to use internal Vref_out of 5V BUT internal Ref_Amp (see below) cannot drive the output to 5V while using 3.3V supply voltage, V+.
If you absolutely need a Vref of 5V, you must increase the supply voltage above 6.25V - see below.
Art,
I believe the issue is caused by the violation of the INA125 output voltage range. On a single 3V supply, the linear output range is from 0.3V to 1.8V (see datasheet table below) but with the output referenced to ground your customer attempts to drive it too close to negative rail.
For 1mV-4mV differential input voltage, in Gain of 64, the output would have to swing between 64mV to 256mV, which is below the minimum specified of 300mV.
To solve the issue you may either:
1. Reference the output voltage above the ground by connect IAref (pin 5) to voltage of 250mV or higher
2. Use a negative supply (pin 3) of at least -250mV
3. Power negative supply (pin 3) with LM7705 (-230mV negative charge pump generator)
4. Increase the gain to at least 300 (RG=200phm) so for Vdiff=1mV Vout>300mV
Hi Art,
what of the proposed changes did the customer carry out? Can the customer show an updated schematic?
It would be a good idea to take higher supply voltages and especially bipolar supply voltages (!) at the beginning and to reduce them to the intended values later. I say this, because sometimes the load cell show so high offset voltage that with small supply voltages or even single supply voltage (!) the output signal can disappear in the output saturation and no one might understand what's going on 
So again, get the circuit running with recommended standard (bipolar!) supply voltages and - if everything is working fine - reduce the supply voltages to the wished level in a second step.
Divide and conquer 
Kai
Art,
For the circuit to work properly, customer needs to implement only one of the recommendations above and NOT all of them:
1. Reference the output voltage above the ground by connect IAref (pin 5) to voltage of 250mV or higher
IAref must be a low impedance source and thus customer may NOT directly connect voltage from a resistive divider to pin 5 because as the current flows in or out of voltage divider this will cause error (see Fig 2 below) - thus, the 300mV output must be buffered before connecting it to pin 5 (see Fig 3). Alternatively, customer may use REF3312 (1.25V reference voltage) to drive pin 5 (see Fig 4).
4. Increase the gain to at least 300 (RG=200phm) so for Vdiff=1mV Vout>300mV
Increasing the gain to 16 for Vdiff=1mV results in the output of: Vout=Vref+1mV*16 =16mV (for Vref=0). This is way below the minimum of 300mV linear range of the INA125 output stage. Thus I recommended gain of 304 so the circuit works with pin 5 being grounded (see Fig 5).
However, if you drive IAref to 300mV you may use any gain you wish - just rememeber to use a buffer for 300mV voltage divider, or reference voltage like REF3312, to drive IAref pin 5 (see Fig 6).
