Calibrating the output voltage
I think you will find chopper stabilised amplifiers with low enough offsets. 

And if you can guarantee linearity in the analog stage better than 0.025% (1 part in 4000) you can calibrate at 0, +1V, -1V and the rest will follow. 

I think the tricky bits will be :


sourcing exactly 1V within 0.25mv and keeping it there across temperature.
verifying that all sources of error combined do not exceed your error budget.