Actually it is not known that Forrest gave an order to shoot the black men like dogs although the Confederate Sgt who wrote that is quoted often.
Perhaps he did but here is a compilation of other quotes from people at the Fort Pillow massacre , men from both sides and both black and white.
If Forrest is the one who ordered the shooting like dogs he also ordered his men to stop and he got between his men and Union soldiers.
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Survivors of Fort Pillow quotes
Two days after the battle
Achilles V. Clark, a Confederate soldier, wrote his sister a letter about what he witnessed in the fight. Clark stated: “ ‘The poor deluded negroes would run up to our men fall upon their knees and with uplifted hands scream for mercy but they were ordered to their feet and then shot down. The white men fared but little better. Their fort turned out to be a great slaughter pen—blood human blood stood about in pools and brains could hav been gathered up in any quantity. I with several others tried to stop the butchery and at one point had partially succeeded—but Gen. Forrest ordered them shot down like dogs and the carnage continued.’ ”
Confederate Samuel Caldwell said the Union soldiers had refused to surrender “which incensed our men & if General Forrest had not run between our men and the Yanks with his pistol and saber drawn not a man would have been spared.”
Private Ellis Falls, a black Union soldier, said that Forrest ordered the Confederates to “quit fighting.”
Private Major Williams, a Union soldier, said he heard one Confederate officer shout that the blacks should be killed and then another Confederate officer contradicted him and said that Forrest had ordered them to capture the blacks and return them to their masters.
One captured Federal,
Charles Fitch, ran up to Forrest and asked for protection. When Fitch said he was from Iowa, Forrest then said: “What in hell are you down here for? I have a great mind to have you killed for being down here.” Forrest then told one of his soldiers to keep Fitch safe.
First Sergeant Wilbur Gaylord of the 6th U.S. Heavy Artillery, a black soldier, was wounded as the Confederates came over the wall. He offered to surrender and was taken prisoner. But at the same time he saw a Confederate soldier “shoot down three black men who were begging for their lives, and who had surrendered.”
http://www.tn4me.org/sapage.cfm/sa_id/28/era_id/5/major_id/5/minor_id/4/a_id/24
Forrest is not a sympathetic figure and likely did vile things in war, since most do, but the above quotes cast doubt on whether he is the one who gave that actual order 11 quoted above.