I don’t quite understand, contemporary drafts? What he said in his speech at Gettysburg is what he said at Gettysburg, it can’t be changed.
Do you have the video or audio of Lincoln giving the Address? If so we can settle this right now.
Otherwise, we simply don’t know what specifically was said. We have the two drafts, and the three post-speech versions. I don’t think there was a transcription: The speech was so short people at the event didn’t even realize the President was speaking until he was almost finished.
However, having said that, the Bliss copy - with “Under God” is the accepted official version. So Obama’s just being an a**hole in using the draft text.
By contemporary I meant drafts made at the time of the speech or soon after. Maybe not the best word. Anyway, Lincoln answered various requests for copies of the speech, and those included it, while the drafts made before the speech don't. Check out the link I posted for more details.
Anyway, my point in mentioning it is not to defend Obama, but to point out that there could be another explanation. Before people get to worked up it makes sense to find out which version Obama was given. If it turned out Burns used one of the other versions making a big deal accusing Obama would just make conservatives look foolish.