Suppose $T$ is a $\lambda$ categorical theory in a language with a unary predicate $P$, and let $\lambda \geq L(T)$. Let $M, N$ be models of $T$ of cardinality $\lambda$, and $M \prec N$ (proper) such that $P^M = P^N$ and $P^M$ infinite.
Then prove that there is $M^* \vDash T$ such that $|P^{M^*}| = \lambda$ and $|M^*| = \lambda^+$
$T$ is complete thanks to categoricity, and thus $M, N$ are isomorphic. To be honest I don't have much more than that. I'm not sure what $M, N$ have to do with it all. I was thinking that the existence of a model with an infinite predicate could let me apply compactness but that's clearly not it given all the info about $P^M = P^N$ and so on. The fact that we have both an elementary extension and an isomorphism doesn't mean much to me either. There seems to be a lot of parts here and I'm not sure what role most of them play.
Note that since $M$ is of cardinality $\lambda$ and $T$ is $\lambda$-categorical, it follows that $M$ is $\lambda$-saturated. In particular, $P^M$ has cardinality $\lambda$.
Thus, it suffices to find a strictly increasing elementary chain $(M_\alpha)_{\alpha<\lambda^+}$ such that $M_0=M$ and for each $\alpha$, we have $P(M_\alpha)=P(M)$ and $\lvert M_\alpha\rvert=\lambda$. If I'm not mistaken, this chain can be constructed using the hypothesis, by straightforward induction. Categoricity is used to ensure that you still have an isomorphic structure at the limit steps.