Jefferson County Public Schools has sworn officers that patrol the schools after hours, but are not allowed to carry weapons per school board policy.
This is in addition to the (relatively new; 10 years or so) SROs; I don't think every school even at the middle and high school level have SROs, though.
They go through the full regular police academy like any other LEO in the state, including using weapons in the academy, but don't have them when they start working. Makes a lot of sense, I know.
My idea for here, at least, would be to arm those folks that patrol, and have armed school "special police", basically security officers sworn as "Special Law Enforcement Officers" under state law, to be in the schools, at least one for EVERY school, during the hours they're occupied.