The writer is saying that it's a good thing that Lincoln was familiar with the King James version, because otherwise, his language in the Gettysburg Address would have lost much of its spirit; the writer argues that Webster went too far in modernizing the poetic language of the King James Bible, in effect taking away its energy and beauty.
By looking at the context, we can see that "Lincoln, born in 1809" would not have been raised hearing "Webster’s “Holy Bible … with Amendments of the Language” [which] appeared in 1833". The writer uses the unusual construction "Lincoln sucked the King James" in reference to the style of the King James version itself, and the idea that the book could figuratively "give suck", as discussed earlier in the piece:
There is a great deal of sucking in King James. (It was often said, in the seventeenth century, that the good book itself “gave suck.” “Milk for Babes, Drawn out of the Breasts of both Testaments” was the title of a popular catechism.) Not in Webster’s. Job was “nursed.” Other infants are “nourished.”
Similarly, the writer chooses stinkethéd to emphasize the choice of language from the King James version:
When Lazarus has been dead for four days, in the King James (John 11:39), “he stinketh.” In Webster, “his body is offensive.”
The writer is saying that the Gettysburg Address would have stunk if Lincoln had been influenced by Webster, perhaps meaning us to associate this both with the current meaning of inferiority (the address would have been less than incredible) and the reference to death earlier in the piece (the language would have been lifeless).
Stinkethéd is probably a past tense created by the writer. (The accent on the -éd indicates that this syllable would be pronounced on its own, as might be needed to fit the rhythm of music or poetry, something like stink-eth-ed instead of a two-syllable stink-eth'd). Invented or not, this is a familiar pattern when we are meant to recall an archaic style. This Catherine Tate clip shows an exaggerated example that references Shakespeare:
Looketh at my face. Looketh at my face. Ist this a bovveréd face thou seest before thee?