INA125 single supply small weight problems
It's likely you're being hit by a couple of nonideal characteristics of opamps: offset voltage, and output swing.

Every opamp has an offset voltage, meaning that it will output slightly above or below the desired setpoint voltage. Since your output is ground-referenced, if the offset is negative, the INA125's output won't rise above ground until you've exceeded its output offset voltage. This is exacerbated by any inaccuracies in your resistor bridge, too. You can observe this if you graph the opamp's output in response to input and observe where the graph crosses the Y axis.

Additionally, no opamp can really output exactly ground. If we look at INA125's datasheet, we can see that typical negative output swing is V- + 0.15 volts - so any output that should be less than 0.15V will instead be about 0.15V.

The solution for both of these is to trim the output using the VREF terminal, as described in the "Offset trimming" section. Though the example schematic there uses a bipolar supply, you can adapt it to a single supply with a simple voltage divider as input to the buffer amp. If you're feeding the output of this into an ADC, provide it with the VREF signal as well, and do a differential conversion to get an accurate result.