I don't have the final vote figures in front of me right now, but it is my impression that the vote wasn't that close for Lincoln to actually need the votes of the soldiers. It is evident that Lincoln may have thought that he was GOING to need that voting block during the summer of 1864, because he and Grant certainly made many false reports on the actual casualities and used the "Black Troops" to keep "White" casualities from impacting his re-election bid in 1864. McClellan was running on a "Peace Platform" so extremely high casualities reports from those battles in 1864 would have effected that Campaign.
As a side point I don't believe that we have any records which show that the over 180,000 plus Black soldiers in the Union Army voted. Or did They? Technically, they had no legal status to vote and a large part of the Union army was made up of immigrants at tha stage of the war also. So I wonder how well that voting block vote was verified? probably not at all.