Can all structures have "lots of compactness" with respect to expansions by constants? This is one of those "surely not ..." questions that I embarrassingly can't answer at the moment.
Given a structure $\mathcal{A}$ in a language $\Sigma$ and infinite cardinals $\kappa<\lambda$, say that $\mathcal{A}$ is $(\kappa,\lambda)$-compact iff whenever $\Sigma'$ is an expansion of $\Sigma$ by constant symbols, $T$ is a $\Sigma'$-theory of cardinality $<\lambda$, and for every $S\subseteq T$ of cardinality $<\kappa$ there is a $\mathcal{B}_S\models S$ with $\mathcal{B}_S\upharpoonright \Sigma=\mathcal{A}$, then there is a $\mathcal{B}\models T$ with $\mathcal{B}\upharpoonright\Sigma=\mathcal{A}$.  For example, $\mathcal{A}$ is never $(\vert\mathcal{A}\vert^+,\vert\mathcal{A}\vert^{++}$)-compact, but if $\vert\mathcal{A}\vert<\kappa$ and $\kappa$ is measurable then $\mathcal{A}$-satisfiability is $(\kappa,\kappa^+)$-compact.
This is a notion I've thought about a bit in the specific case of $\mathcal{R}=(\mathbb{R};+,\times)$ (see here or here), but I'm also interested in it more generally. Very little about it is obvious to me. As a starting point, I'd like to know if the following "compactness at many inaccessibles" phenomenon is consistent:

${\bf (CMI) } \quad$ For every structure $\mathcal{A}$ and every cardinal $\kappa_0$ there are cardinals $\kappa_0\le \kappa<\lambda$ such that $\lambda$ is inaccessible and $\mathcal{A}$ is $(\kappa,\lambda)$-compact.

The inaccessibility requirement on $\lambda$ ensures that $\kappa$ is "vanishingly small" compared to $\lambda$, in which case $(\kappa,\lambda)$-compactness seems implausibly strong, and I don't offhand know of any nontrivial example even of $(\kappa,\kappa^{++})$-compactness of a structure. However, at the moment I can't rule (CMI) out. In fact, at the moment I can't even prove that the failure of (CMI) is consistent with $\mathsf{ZFC}$ + a proper class of inaccessibles.
 A: To move this off the unanswered queue, I'm turning Harry West's comment above into an answer. If he posts an answer of his own, I'll delete this one and accept it instead; meanwhile, I've made this CW to avoid reputation gain.
The answer is yes, at least if we're willing to accept rather strong large cardinal axioms as consistent with $\mathsf{ZFC}$: the principle CMI follows for example from the existence of a proper class of strongly compact cardinals. This is because every structure can be described up to isomorphism by a single $\mathcal{L}_{\mu,\mu}$-sentence for some sufficiently large $\mu$ (the successor of the cardinality of the structure is always enough, but sometimes we can do better).
So suppose $\mathcal{A}$ is a $\Sigma$-structure and $\kappa_0$ is a cardinal. Let $\theta$ be a strongly compact cardinal $>\vert\mathcal{A}\vert+\kappa_0$ and suppose $T$ is a theory of arbitrary cardinality, in an expansion $\Sigma'$ of $\Sigma$ by constant symbols, every size-$<\theta$ subtheory of which has a model with $\Sigma$-reduct $\mathcal{A}$. Let $\varphi$ be a $\mathcal{L}_{\mu,\mu}$-sentence characterizing $\mathcal{A}$ up to isomorphism, with $\mu<\theta$; this exists since $\vert\mathcal{A}\vert<\theta$. Now just apply the strong compactness of $\theta$ to the theory $T\cup\{\varphi\}$.
