This doesn't answer directly the question of Lincoln's interaction with USCTs, but it does describe his visit to Richmond, roughly within the period covered in the film, and meeting with freemen there.
From the Atlantic, June 1865:
He entered the city unheralded; six sailors, armed with carbines, stepped upon the shore, followed by the President, who held his little son by the hand, and Admiral Porter; the officers followed, and six more sailors brought up the rear. The writer of this article was there upon the spot, and, joining the party, became an observer of the memorable event.
There were forty or fifty freedmen, who had been sole possessors of themselves for twenty-four hours, at work on the bank of the canal, securing some floating timber, under the direction of a Lieutenant. Somehow they obtained the information that the man who was head and shoulders taller than all others around him, with features large and irregular, with a mild eye and pleasant countenance, was President Lincoln.
"God bless you, Sah!" said one, taking off his cap and bowing very low.
"Hurrah! hurrah! President Linkum hab come!" was the shout which rang through the street.
The Lieutenant found himself without a command. What cared those freedmen, fresh from the house of bondage, for floating timber or military commands? Their deliverer had come,—he who, next to the Lord Jesus, was their best friend! It was not an hurrah that they gave, but a wild, jubilant cry of inexpressible joy.
They gathered round the President, ran ahead, hovered upon the flanks of the little company, and hung like a dark cloud upon the rear. Men, women, and children joined the constantly increasing throng. They came from all the by-streets, running in breathless haste, shouting and hallooing and dancing with delight. The men threw up their hats, the women waved their bonnets and handkerchiefs, clapped their hands, and sang, "Glory to God! glory! glory! glory! "—rendering all the praise to God, who had heard their wailings in the past, their moanings for wives, husbands, children, and friends sold out of their sight, had given them freedom, and, after long years of waiting, had permitted them thus unexpectedly to behold the face of their great benefactor.
"I thank you, dear Jesus, that I behold President Linkum!" was the exclamation of a woman who stood upon the threshold of her humble home, and with streaming eyes and clasped hands gave thanks aloud to the Saviour of men.
Another, more demonstrative in her joy, was jumping and striking her hands with all her might, crying, —"Bless de Lord! Bless de Lord! Bless de Lord!" as if there could be no end of her thanksgiving. . . .
Abraham Lincoln was walking their streets; and, worst of all, that plain, honest-hearted man was recognizing the "******s" as human beings by returning their salutations! The walk was long, and the President halted a moment to rest. "May de good Lord bless you, President Linkum!" said an old negro, removing his hat, and bowing with tears of joy rolling down his cheeks. The President removed his own hat, and bowed in silence; but it was a bow which upset the forms, laws, customs, and ceremonies of centuries. It was a death-shock to chivalry, and a mortal wound to caste. Recognize a ******! Faugh! A woman in an adjoining house beheld it, and turned from the scene in unspeakable disgust. There were men in the crowd who had daggers in their eyes; but the chosen assassin was not there, the hour for the damning work had not come, and that great-hearted man passed on to the executive mansion of the late Confederacy.
That last graf just kills.
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Image: Freedmen by the canal, Richmond, April 1865. Library of Congress.
Bringing the