= Solution
\b[For an arbitrary ground model, this alternative needs an additional arithmetic hypothesis.] The requested countable-cover property already prevents any collapse of an infinite ground-model cardinal. If a cardinal $\theta$ were collapsed, there would be a surjection $f:\alpha\to\theta$ with $|\alpha|^M<\theta$. The asserted ground-model $F$ would cover $\theta$ by the union of $|\alpha|^M$ countable sets, a ground-model set of size at most $\max(\aleph_0,|\alpha|^M)<\theta$, a contradiction. Thus all infinite cardinals are preserved.
Put $\lambda=(\aleph_3)^M$, which must then remain $\aleph_3$. If the extension has continuum $\lambda$, it has
$$
\lambda^{\aleph_0}=(2^{\aleph_0})^{\aleph_0}=\lambda.
$$
The ground-model functions from $\omega$ into $\lambda$ remain present and their cardinality cannot be collapsed. Necessarily
$$
\boxed{M\models\lambda^{\aleph_0}=\lambda.}
$$
For example, a ground model with continuum larger than $\aleph_3$ cannot satisfy the printed request. This is a genuine missing hypothesis, rather than a forcing construction that works for every $M$.
Under the necessary hypothesis, use <Cohen forcing> to add $\lambda$ reals:
$$
\mathbb P=\operatorname{Fn}(\lambda\times\omega,2,{<}\omega).
$$
The <delta-system lemma> shows that it has the <countable chain condition for forcing>: an uncountable family of finite conditions has an uncountable subfamily whose domains form a delta-system and whose values agree on its root, so any two of that subfamily are compatible. It therefore preserves cardinals. Its $\lambda$ coordinate reals are pairwise distinct by dense disagreement requirements. Conversely each <nice forcing name> for a real uses countably many countable <antichains in a forcing order>, so the number of such names is at most $\lambda^{\aleph_0}=\lambda$. Hence
$$
\boxed{M[G]\models2^{\aleph_0}=\aleph_3.}
$$
For any ordinal-valued function name, choose for each $\xi<\alpha$ a maximal <antichain in a forcing order> deciding its value. The <countable chain condition for forcing> makes the set $F(\xi)$ of possible ordinal values countable in $M$. Pad it with $\omega$ if needed to make it countably infinite. The <possible-values lemma for chain-condition forcing> gives the stronger pointwise covering statement
$$
\boxed{f(\xi)\in F(\xi)\quad(\xi<\alpha),}
$$
which implies the requested range inclusion. When a condition only forces that the name is such a function, make these choices below that condition; it belongs to the generic filter witnessing the actual function.
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