  We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of
$K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's
work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the
Noether-Lefschetz locus in $\mathcal{M}$ is a countable union of special
divisors $\mathcal{M}_d$, where the discriminant $d$ is a positive integer
congruent to $0,2,$ or $4$ modulo 8. We compute the Kodaira dimensions of these
special divisors for all but finitely many discriminants; in particular, we
show that for $d>224$ and for many other small values of $d$, the space
$\mathcal{M}_d$ is a variety of general type.
