SWas it ever possible for racial tensions in the South so bad that a full-out civil war or race war erupts in the former Confederacy in the 1960’s?
No. Not even close. Not remotely close.
The actual Southern armed rebellion in 1860-1861 broke out because at the time, a dominant minority of Southerners believed that the election of Lincoln posed an
existential threat to white Southerners. In 1860, the seven Deep South States were 46% slaves. The states of South Carolina and Mississippi were majority slaves, as were large areas in the other five. White Southerners feared what the blacks would do if the slavery system broke down or was subverted. The spectre of Haiti (where rebel slaves had largely wiped out the white population) was ever-present. John Brown's expedition at Harpers Ferry was seeming evidence that Northern abolitionists would seek to incite such violence. Beyond this was the obvious point that abolition of slavery would cost Southerners billions of $.
Under these conditions, the militant secessionist plurality in the South managed to stampede several states into secession. (South Carolina had been hot to trot for years, but Georgia, Alabama, and Louisiana were reluctant. Texas had to override Governor Sam Houston.)
Also, of course, since secessionist rebellion had never been tried before, the secessionists could argue that a) the North wouldn't fight, and b) if they did, Southerners would beat them.
In the 1960s? There was no panic. No one was afraid the blacks would rise up and cut white throats. (And demographics had shifted - no states were majority black any more.) There were some businesses who stood to lose from the end of Jim Crow, but not an entire region-wide class. And in fact the South had retreated considerably on race issues. Lynching (long defended as a cornerstone of white supremacy) was essentially over. (There were at least 50 lynchings each year through 1921 (except 1911).There were 25 lynchings in the 19 years after World War II. The last officially recorded lynchings were the murder of three civil rights workers in 1964.)
Another demographic shift was that several "Southern" were no longer homogenously "Southern". For instance, the southern half of Florida was mostly northern migrants. Texas had moved from being "Southern" to being equally "Western".
Finally, and most importantly, after the War, Southerners dared not fight the Federal government. In the era from after Reconstruction to 1948, white Southerners defended white supremacy by influencing the Federal government, mainly through the Democratic Party. Southern Senators employed their seniority to block civil rights legislation (and get lots of Federal pork). When the scheme broke down after 1948, all they could do was bluster. The gasconading and flag display was just that - display.
When the Supreme Court struck down their prized rules, and Presidents enforced those decisions with US Army troops... Southerners talked a lot, but they didn't care enough to risk actual fighting. George Wallace promised to "stand in the schoolhouse door" - but he got out of the way when Federal marshal walked up escorting black studens.