I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ at $\aleph_\omega$ is to start with a suitable model with a large cardinal $\kappa$ satisfying $2^\kappa=\kappa^{++}$, and then force with "Prikry forcing with interleaved collapses".
A more naive approach would be ordinary Prikry forcing followed by a (full support) product or iteration of Levy collapses. Is there absolutely zero chance for this to work? I know there are subtleties, e.g., we cannot expect all cardinals above $\kappa$ to be preserved if we start with a large $2^\kappa$, due to Shelah's bound $2^{\aleph_\omega}<\aleph_{\omega_4}$.
To be concrete, say $\kappa$ is a singular strong limit cardinal with cofinality $\omega$ and $2^\kappa=\kappa^{++}$. Suppose $\kappa_n$ is an increasing sequence of regular cardinals with limit $\kappa$. Does something like $\prod_n\mathrm{Col}(\kappa_n^{++},<\kappa_{n+1})$ necessarily collapse $\kappa^{++}$? It does preserve $\kappa^+$ by usual arguments. If the partial order $(\prod_{n}\kappa_n^{++},<^*)$ has a cofinal subset of size $\kappa^+$ (equivalently a scale of length $\kappa^+$), then $\kappa^{++}$ will be collapsed by a density argument; $f<^*g$ means $\{n\in\omega:f(n)\geq g(n)\}$ is finite. Jech's paper On the cofinality of countable products of cardinal numbers contains much information about cofinalities of countable products. In particular, if we are in a Prikry generic extension $V[G]$ using some measure $U$, $j:V\rightarrow M$ is the corresponding elementary embedding, and $(\kappa_n:n<\omega)$ is the Prikry sequence, then the cofinalities of $\prod_n\kappa_n$ and $\prod_n\kappa_n^+$ are respectively $\mathrm{cf}^Vj(\kappa)$ and $\kappa^+$. But I'm not sure how to compute the cofinality of $\prod_{n}\kappa_n^{++}$; it seems to be $\kappa^{++}$ in case $(\kappa^{++})^M=\kappa^{++}$; that would mean the above attempt to show $\kappa^{++}$ is collapsed doesn't work. There is also the question of what if we use some $(\kappa_n:n<\omega)$ that doesn't come from the Prikry generic sequence.
