First, Selma's not the only big-history picture to deliberately alter history for drama; famously or infamously, Steven Spielberg de-unanimized Connecticut's Congressional vote on the 13th Amendment.
That said, that was different than this; it didn't fundamentally change the perception of a main character in the movie.
By claiming LBJ sicced J. Edgar Hoover on Martin Luther King, though, Selma does exactly that.
First, it's pretty clear that LBJ had no advance knowledge of Hoover's "tomcat" letter to King, in which he attempts to force King to drop his civil rights work, perhaps even commit suicide. It's not clear how long after it was sent, in fact, that LBJ knew about it.
Second, if anybody sicced Hoover on King, it was the Kennedys.
Third, since LBJ had as much trouble keeping his pecker in his pants as did Jack Kennedy, Hoover had the drop on him. If LBJ had known in advance about the "tomcat" letter, he might have acquiesced. Not that that would have been good, it's just the reality. (Sidebar: Presuming the homosexuality rumors about Hoover are true [deliberately not using the word "gay"], he obviously kept it buried through a mix of discretion and of having the goods on anybody who seriously thought about trying to leverage him.)
At the same time, the movie's handling of this is part of how it gets King wrong.
King didn't let this letter stop him from involvement in the second Selma march. Rather, that march didn't have a green light from a federal judge, so King wouldn't march, beyond the famous "turnaround," discussed in Wikipedia's overview of Selma 1965. This had nothing to do with Hoover's letter, and everything to do with details of King's nonviolence strategy, and not breaking the law when breaking the law would have hurt the movement.
And, thus, one error becomes compounded into two. Or more.
In turn, that seems to be part of larger problems with Selma the movie's treatment of King, and of King vis-a-vis other parts of the civil rights movement.
Basically, the movie underplays the level of distrust between King, and the likes of James Forman and others from SNCC, from what I understand. So, we get a sanitized view of the civil rights movement, along with a Manichean view of LBJ vs. MLK.
I also find this response, noted in the Wikipedia article about the movie, disturbing:
Director (Ava) DuVernay and U.S. Representative John Lewis (whom she portrays when a young man) responded separately that the film Selma is a work of art about the people of Selma, not a documentary. DuVernay said in an interview that she did not see herself as "a custodian of anyone's legacy". In response to criticisms that she rewrote history to portray her own agenda, DuVernay said that the movie is "not a documentary. I'm not a historian. I'm a storyteller". Lewis wrote in an op-ed for The Los Angeles Times: "We do not demand completeness of other historical dramas, so why is it required of this film?"
And, DuVernay's own comments indicate that she seems to have a postmodernist view of issues of historical accuracy, with her "accurate to whom?" comment. Besides that, she admits that it was about drama, and her own drama, when she says elsewhere that it's about "what I feel about LBJ."
Is it a bad movie? I don't think so.
Could it have been an even better movie, within the budgetary, independent film constraints? Yes. Both more historically accurate AND, I think, more dramatic.
And, claiming that its critics are "smearing" the movie? Comes close to the social justice warrior POV, to me.