True. Equivalently, the family must be chosen so that $C$ does not admit a first order thickening over $\textrm{Spec}\,\mathbb{C}[[t]]$, i.e. should be transverse to the locus of K3s with a $-2$-curve.

Unfortunately, I don't really have the expertise to understand the papers I found when searching those keywords. By simple function do you mean a sum of step functions or is there another meaning to the word "simple"? Does the theta correspondence produce (mock, quasi, etc) modular forms (functions, etc)? I only care about the coefficients of $q^n$ for $n>0$ so one could modify the above series to sum over all orbits of vectors, or add a Laurent tail. I don't know if that would simplify things.