= Solution
Put $\kappa=\aleph_2^M$ and let $F=\bigcup G$. Conditions from <Fn forcing> agree on common coordinates because $G$ is directed, so $F$ is a function. For each $(\xi,n)\in\kappa\times\omega$, the ground-model <set>
$$
D_{\xi,n}=\{p:(\xi,n)\in\operatorname{dom}p\}
$$
is dense: add a value at that coordinate if necessary. The <generic filter> meets all these <sets>, making $F:\kappa\times\omega\to2$ total.
For distinct $\xi,\eta<\kappa$, the ground-model <set>
$$
D_{\xi,\eta}=\{p:\exists n\ ((\xi,n),(\eta,n)\in\operatorname{dom}p\text{ and }p(\xi,n)\ne p(\eta,n))\}
$$
is dense. Choose $n$ fresh for both rows of a finite condition and assign $0$ and $1$ there. Genericity therefore makes the rows pairwise different. The <generic coordinate reals for finite-function forcing> are $c_\xi=\{n\in\omega:F(\xi,n)=1\}$, and in the <generic extension>
$$
\boxed{\xi\longmapsto c_\xi\text{ is an injection }\aleph_2^M\hookrightarrow\mathcal P^{M[G]}(\omega).}
$$
The family is a <set> in $M[G]$, for example by collecting the ground-model names $\dot c_\xi=\{(\check n,p):p(\xi,n)=1\}$ and evaluating the corresponding name for their indexed family. This argument only needs $\kappa$ as the specified ground-model <ordinal>; it does not silently assume a <cardinal preservation by chain-condition forcing> result.
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