Is it sure the nonlinearity is from the GLN and not from the DMMs ?
A simple test to do for this would be to repeat a test with the DMM reversed. This would at least change some of the DMM caused INL, especially the even powers.
The rather wide curve from the 6500 looks a bit like this could be an effect of the meter, maybe an effect of a limited number of digits or internal resolution limitiation.
I would expect the time base errors to be very small, so no worries there.
Another possible error source may be DA of capacitors in the filter. There is a large 10 µF MKS cap on the board. Is that doing ouput filtering (could cause trouble) or is it more reference filtering (would be OK) ?
10 µV are still quite a bit and usually more than to expect from thermal EMF, and the loading error to an OP like OPA140. For the OP error I would be more afraid of output cross over, that may cause some 1 µV step when the polarity of the current changes, though still a relatively small error.