Let $F_{g,n}$ be the moduli space of $n$-pointed K3 surfaces of genus $g$ with at worst rational double points. We prove that the ring of pluricanonical forms on $F_{g,n}$ is isomorphic to the ring of orthogonal modular forms of weight divisible by $19+n$, twisted by the determinant character and with vanishing condition at the $(-2)$-Heegner divisor. This maps canonical forms on a smooth projective model to cusp forms. Then we use Borcherds products to find a lower bound of n where $F_{g,n}$ has nonnegative Kodaira dimension, and compare this with an upper bound where $F_{g,n}$ is unirational or uniruled using classical and Mukai models in $g<21$. In some cases, this reveals the exact transition point of Kodaira dimension. When n is sufficiently large, the Kodaira dimension stabilizes to $19$.