I have a theory with finitely many relations, and would like to find a model of it with continuum-many 1-types realized, and one 2-type omitted. Is there a version of the Omitting Types Theorem that would help me in this case? The proof in Marker's "Model Theory: An Introduction" explicitly uses countability of the language, so I can't for instance introduce continuum many constant symbols for the types I want to realize.

Specifically, I'm working with the weak monadic second-order theory of 1 successor (treated as a two-typed first order structure). I'm curious if there is a nonstandard model whose "first order part" is $\omega + \zeta$ (thus omitting the 2-type of having two points infinitely far from each other and infinitely far from 0), but whose "second order part" $B$ satisfies: $\{X \cap \omega | X \in B\} = \mathcal{P}(\omega)$. That is, for every subset of the natural numbers, there is a "second order element" in the nonstandard model containing exactly those natural numbers (and then some collection of nonstandard elements as well).