Hello, escoville. Thanks for doing such a thoughtful analysis. But I'm not sure what your verdict was.
I hope we can agree that 'I am father.' is impossible.
I hope we can also agree that 'I am a father of two' is possible.
Yes, we do agree about that.
The case of Abraham Lincoln is clearly an instance of (a), as are the two cases in Cowper's poem.
Regarding your analysis: If we assume that the Lincoln sentence and the Cowper poem show an "omission" of something, then in the Lincoln sentence it is a definite article that was omitted. (Thomas was "the"--not "a"--father of Abe.) However, I see no reason to make the same assumption about the Cowper poem, which could mean either that he is "the" lord of fowl and brutes or "a" lord of fowl and brutes. (Maybe he has a brother who is also such a lord--a brother who lives in another area, perhaps somewhere in Wisconsin.)
I don't know whether the Lincoln and Cowper examples represent the "omission" of anything, or whether, instead, they simply are good English in any event (and in their own right).
I am happy to take your analysis as correct, if the upshot is that such things ("He is father of two (sons)"; "Thomas, father of Abraham"; "I am monarch of all I survey") are deemed to be good, correct English. I say they are good, correct English.