In the poem "My Last Duchess" by Robert Browning, the narrator is omniscent only in the fact that he knew what really went on with her murder, he knew how it happened, when and by whom, and he knows why he is telling his visitor the details that he is.  So, he is "all-knowing" or omniscent in the sense that he has a motive for telling his tale to the representative of the new wife, and he knows the real story behind his first wife's death.  So, of the two options you have provided above, I think that omniscent is the better choice--he isn't a very objective speaker; he has been emotionally involved in the tale, has a motive and underlying agenda, and so chooses and presents his facts depending on who is around.  That describes someone as subjective.  So, go with omniscent if those are your only two choices, but I still don't like it.

A better description for the narrator is to describe him as an unreliable narrator.  Telling the story is a man who "gave commands" to have his wife murdered.  How much can we really trust him?  And, he obviously has some pretty glaring weaknesses; his wounded ego and pride led to the murder, and we never get the wife's perspective.  All we get are his descriptions of how she was

"too soon made glad, /Too easily impressed: she liked whate'er /She looked on, and her looks went everywhere."

We never get to hear her point of view.  Did she really disrespect his "nine-hundred-years-old name" as he claimed she did?  Did she really flirst and blush at everything or was that his wounded ego imagining things?  His perspective of things is incomplete, and biased, so he is not a reliable narrator, not one we can trust to present a fair and balanced view of the events.

I hope that those thoughts help to get you thinking a bit; good luck!

"My Last Duchess" is a narrative poem by Robert Browning that is structured as a dramatic monolog. The speaker is the cold and arrogant Duke Ferrara who escorts a visitor through his castle, pausing before a portrait of the Duke's previous wife who, we learn, has died. The poem's point of view, therefore, is that of a first-person narrator. The story is limited by what he chooses to disclose about what he knows of her.

The Duke is not an objective narrator. As he speaks of the Duchess, he reveals himself to have been cruel, dismissive, and domineering toward her. When he speaks of her, he reveals her personality, as well, but the qualities in the Duchess that he scorns are those of an open and loving person. The gentle Duchess would have been valued by anyone who lacked the Duke's arrogance and need to dominate.