A question about non-linearity of an in-amp gain
I followed ThePhoton's suggestion in the comments.

Here is how to solve this:

First take offset by tying inputs together and to the GND and measure the output voltage Vout. In this case -39.3478mV as shown below:



So this should be subtracted from any output voltage after any measurement. And when I do that gain looks 800 constant:



*I suppose if a strain_gage is mounted and fixed, an offset measurement can be taken before any moving measurement(without the procedure above). Voffset is the output voltage when the inputs are tied to GND.

Imagine when mounted the offset stress of the gage is X amount(Newton) and the voltage is Vxo = V1+Voffset . If the stress is varied ΔX amount there will be increase in output voltage as ΔV and the output will be V2=(V1+Voffset)+ΔV.
Now V2-V1 = ΔV. So the amplifier's Voffset is included in the first offset already.