The idea of the test with the resistor chain is to measure INL at a few points. As ideally INL is rather small, one usually has to push it hard to avoid other errors. Offsets in the 1-2 µV range, e.g. from hand warm banana plugs might already be a problem.
A second type of difficulty can be settling of the meter and reference drift. A fast measurement sequence might see settling effects, while a slow sequence would see more drift effects. One could do a separate settling test with sending the data to the PC.
The different values (1.3 µV) already at 1 V and thus only the first resistor kind of indicates a problem - either with drift or quite a lot of offset. How good do the measurements of the resistor voltages match initial and at the end ?
One should be able to have both the positive and negative side in one curve (with 0 in the middle) - the voltages over the single resistors can be measured with one sign and than give a continuous sequence through 0.
With the PC interface one should be able to check for input bias at a few more voltages (e.g. with the drift seem with a good quality (PS or PP) capacitor in the 1 nF range). Alternatively one could do the test with and without an extra series resistor. Even if the input bias is in specs (e.g. < 100 pA), one might have to apply corrections, as the source impedance changes with the resistor chain.
As the meter does not use a pre-charge phase, there is a chance that there is some "charge injection" from the MUX changing between 0 and input. If this changes with voltage (which is to be expected) and input impedance this could cause some extra errors. It might be worth doing a test with a positive voltage (e.g. 5 V, 7 V or similar), comparing a few different series resistors (e.g. 0 Ohms, 100 K , 1 M) - if there is a significant difference, this would indicate an input related problem. Capacitance at the input (e.g. C008) could make a difference too, as the switching includes dynamic effects. There is some chance that switching can be different when going up or down in voltage so this could cause a difference between positive and negative voltages.
One might be able to see switching artifacts with the scope at the input.