The FSK doesn't have a direct legal mandate to decide which movies may be shown on quiet holidays. Nevertheless, it's categorization is generally respected.
The FSK is a voluntary self-censorship institution by the German movie industry, which does have a legal mandate for the purpose of child and youth protection per § 14 JuSchG. It primarily provides an age rating for every submitted movie.
In § 28 of its charter, the FSK has tasked itself with also deciding suitability for quiet holidays upon request, noting that these holidays have additional protections under various state laws.
For example, the Bavarian Art 3 FTG outlaws public entertainment events unless the “serious nature” of these special holidays is maintained.
What does and doesn't maintain the serious nature depends on the social context. The classification of the FSK has become substantially more liberal over time. Whereas almost any children movie would have previously been forbidden, it's now largely horror flicks.
It is possible that a movie theatre could show a movie without suitable FSK classification. However, the theatre would do so at its own risk, and would likely get sued by religious/moralistic groups.