TLDR: attempts to remove that section were defeated at the polls (and in the Alabama Supreme court) in no small part due to arguments/judges of the Christian right, despite some Republican support for repealing the section.
A Salon article, although it probably has some (liberal) biases, nonethelss offers some insight into why the 2004 repeal attempt of section 256 (hat tip to @barbacue for finding it) was narrowly defeated:
In 2004, Roy Moore was looking for something new to do. In November of the previous year, he had been thrown out of the Alabama Supreme Court, where he was the chief justice, for installing a 5,000-pound monument of the 10 Commandments in the center of the courthouse rotunda. This was a clear violation of the First Amendment -- “To this, the Establishment Clause says no,” U.S. District Judge Myron Thompson ruled -- but Moore was defiant. After years of litigation and political controversy, the monument was eventually removed and Moore lost his job. But the incident turned him into a Christian-right celebrity, drawing supportive rallies of thousands that featured national conservatives like Alan Keyes and the Rev. Jerry Falwell.
Moore leveraged that fame into his new campaign: fighting Amendment 2. He claimed it was actually a secret back door to increase taxes. "This is the most deceptive piece of legislation I have ever seen, and it is simply a fraud on the people of Alabama,” he told the Associated Press at the time.
At issue was that the amendment would have guaranteed a "right to an education” for all children, something that Brown v. Board had already guaranteed, but that Moore and his allies, especially the state chapter of the Christian Coalition, nonetheless used as a cudgel. Opponents argued that a parent in a poorer county could bring a lawsuit alleging that their children's rights were violated because they received an inferior education to their peers in a richer county, and that a court would then order a tax hike to fix the inequality. Legal experts thought the argument was far-fetched.
As the Washington Post noted after the measure failed, “Employing an argument that was ridiculed by most of the state's newspapers and by legions of legal experts, [Alabama Christian Coalition president John] Giles and others said guaranteeing a right to a public education would have opened a door for ‘rogue’ federal judges to order the state to raise taxes to pay for improvements in its public school system.”
I tried to find independent verification of what Moore said on this, but no luck insofar. But I think it's probably a fair assessment that the Christian right opposed the 2004 repeal attempt... promising their own:
Despite the narrow margin, Christian Coalition chief John Giles declared victory.
"The Christian Coalition of Alabama will work to ensure that reckless trial lawyers and activist judges will not be able to open the floodgates to increase taxes and that private, Christian, parochial and home-school families will be protected," Giles said in a statement on the group's Web site.
"The Christian Coalition will lead the way to remove the racist language in the next election."
Note that the 2004 defeat was so narrow it was subject to a recount (same source as above quote).
Also section 256 was struck down by Alabama's courts (including their supreme one) only to be reinstated once the court's composition changed! From the same source:
In 1993, amid a lawsuit over state education funding, a circuit judge in Montgomery struck down the amendment. The state Supreme Court upheld his ruling in 1997.
But in 2002, during Moore's tenure as chief justice, the court reopened the case and reversed itself.
So it's very likely Moore opposed it in 2004 as well. Actually regarding the 2012-defeated attempt Moore said
Moore told the Associated Press that the amendment was “another attempt to open the door for a court-ordered tax increase without the consent of the people” after they’d defeated the earlier amendment, while [Tom] Parker ran radio ads saying that it would create “a new right to education for citizens of all ages” and warning “liberals will use this to pressure judges into raising your taxes.”
As for background on the latter
The most prominent politician besides Moore battling the amendment was his protege and former staffer, Tom Parker, who was running for the Alabama Supreme Court at the time. During that campaign, Parker spoke at an event celebrating Nathan Bedford Forrest, the Confederate general and Ku Klux Klan leader, hosted by opponents of the civil rights movement, and handed out Confederate battle flags at the funeral of a woman believed to have been the last living widow of a Confederate soldier.