(The original version of this question was much narrower and less natural; but see the edit history if interested.)
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$ (I'll abbreviate $i=\{i_\Sigma:\Sigma$ a finite language$\}$). Using $i$ to conflate sentences with naturals appropriately, we can define an inner model $$M_\mathcal{L}^r:=L[r,\models_\mathcal{L}]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.
Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:
Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$
Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.
A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.
Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"
Note that it is crucial that we ask about an actual measurable cardinal, rather than merely an inner model with a measurable: already the logic $\mathsf{SOL}^{TV}$ mentioned above gives us every projective real, which under large cardinal assumptions (if memory serves) gives rise to very strong inner models.
