Hello Brian,
What you're seeing could be the result of two error parameters. First, as you mentioned, is that the linearity is only specified over a subset of the device's codes, from 0x010 to 0xFFF, which is pretty typical for a DAC. Second, is the unipolar zero error specified as 1 LSB at code 0x010.
The reason that linearity decays and a zero-code error is introduced is usually because of headroom requirements of the internal output amplifier. The amplifier cannot output voltages that are so close to its negative supply rail. A solution would be to provide the device with some negative supply.
You could either operate the device in bipolar mode with a negative supply that is less than approximately -1.2V, where the part transitions to bipolar mode. Alternatively, if you want to stay in unipolar mode, you could try providing a supply that is greater than -1.2V and less than 0V to provide more headroom on the output amplifier's negative rail. A part like the LM7705 might be good enough.
I haven't created and tested such a setup yet myself, but adding -230mV of headroom on the negative rail should certainly help you out.