Does a school syllabus trump state law?

I wouldn't say that it "trumps state law". Indeed, the State of Georgia, either expressly by statute or through the common law, establishes that teachers and school administrators have the authority to create rules and regulations governing the conduct of students that are not themselves unconstitutional as applied to students, although, in general these consequences can't resort to criminal punishments.

Instead, typical punishments include detention, suspension (in school or out of school), expulsion, and adjustment of grades for an assignment or a course. Marks in one's disciplinary record and public shaming, forfeiture of eligibility to participate in school sponsored extra-curricular activities or honors (including marching at graduation), refusing to release transcripts, and historically (but much less so in recent years) corporal punishments such as spanking, have been options for schools to enforce their punishments.

A prohibition on recording in a syllabus certainly wouldn't result in criminal punishments, and probably wouldn't even give rise to civil liability. Depending upon the purpose for which the recording was being used, it is even conceivable that the school's right to punish someone for violating a school rule could be estopped by First Amendment and whistle blower protection law considerations (e.g. if it was used to document harassment and discriminatory conduct for use in sharing with the school board or law enforcement or publishing on radio or TV or an Internet news source).

But, the mere fact that conduct is legal outside a school setting does not mean that a school cannot prohibit and punish that conduct in its own rules.

The closer case, upon which there is more division of legal authority, is under what circumstances a school can legitimately punish conduct away from school, for example, uploading rap lyrics about a teacher to YouTube from home without using any school resources to do so.