= Solution
Use the <countable-condition collapse>
$$
\boxed{\mathbb P=\operatorname{Col}(\omega_1,\kappa)
=\{p:p\text{ is a countable partial function from }\omega_1\text{ to }\kappa\},}
$$
ordered by reverse inclusion: an extension of a function is a stronger condition. All sizes and conditions here are computed in the ground model $M$.
This forcing is <countably closed>: the union of a descending countable sequence is a countable partial function. Thus it adds no countable ordinal sequences and preserves $\omega_1$. For each $\xi<\omega_1$, the conditions whose domains contain $\xi$ are dense; for each $\eta<\kappa$, those whose ranges contain $\eta$ are dense. The union of the <generic filter> is therefore a surjection from $\omega_1$ onto $\kappa$, and $|\kappa|^{M[G]}=\aleph_1$.
Inaccessibility gives $\kappa^{\aleph_0}=\kappa$: every countable sequence is bounded below the regular $\kappa$, and the <strong limit cardinal> property bounds the number of sequences at each bound below $\kappa$. Hence $|\mathbb P|=\kappa$. The forcing satisfies the $\kappa^+$-<chain condition for forcing>, so it preserves every cardinal above $\kappa$. \b[This gives precisely the requested collapse and preservation.] The assertion that $\kappa$ is collapsed to $\aleph_1$ concerns its new cardinality; the ordinal $\kappa$ itself does not change.
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