= Solution
Choose a name $\dot f\in M$ for $f$ and a condition $p\in G$ <forcing> that it is a <function> $\alpha\to\beta$. The <ordinals> $\alpha,\beta$ belong to $M$, by <forcing preserves ordinals>. For each $\zeta<\alpha$, choose in $M$ a maximal <forcing antichain> $A_\zeta$ in the cone above $p$, every member deciding $\dot f(\zeta)$ as an <ordinal> below $\beta$. Conditions deciding an ordinal-valued name are dense, by the <forcing theorem>.
Let $y(\zeta)$ be the set of values decided by members of $A_\zeta$. The chain condition gives $|y(\zeta)|^M<\kappa$, and Choice and Replacement in $M$ assemble all these sets into a <function> $y\in M$ with domain $\alpha$. Since $p\in G$, genericity ensures that $G$ meets each deciding <forcing antichain> above $p$; equivalently enlarge it to a global maximal <forcing antichain> by conditions incompatible with $p$. Its chosen value is the actual $f(\zeta)$. Therefore
$$
\boxed{\operatorname{dom}(y)=\alpha,\quad
M\models|y(\zeta)|<\kappa,\quad
M[G]\models f(\zeta)\in y(\zeta)\text{ for all }\zeta<\alpha.}
$$
This is the <possible-values lemma for chain-condition forcing>. All size bounds in its construction are internal to the ground model.
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