Throughout, by "logic" I mean regular logic (in the sense of Ebbinghaus–Flum–Thomas) whose sentences are coded by elements of $\mathsf{HC}$. Say that $\mathcal{L}$ is Barwise compact iff whenever $\mathbb{A}$ is a countable admissible set and $X\subseteq\mathbb{A}$ is a $\mathbb{A}$-c.e. $\mathcal{L}$-theory each of whose $\mathbb{A}$-finite subtheories is satisfiable, then $X$ is satisfiable. Finally, say that a logic $\mathcal{L}$ has the single-sentence DLS property iff every satisfiable $\mathcal{L}$-sentence has a countable model.
It's easy to show that $\mathcal{L}_{\omega_1,\omega}$ has the single-sentence DLS property (favorite silly proof: apply Mostowski absoluteness to $\mathit{Col}(\omega,\kappa)$ for large enough $\kappa$), and Barwise showed that $\mathcal{L}_{\omega_1,\omega}$ is Barwise compact. Moreover, Lindstrom's characterization (see Väänänen, Lindstrom's Theorem) of first-order logic can be tweaked to show that no logic with the above two properties can be much stronger than $\mathcal{L}_{\omega_1,\omega}$. However, there's still a large gap here, and I'm curious whether anything is known about it. In particular:
Is there an already-introduced logic which is properly stronger than $\mathcal{L}_{\omega_1,\omega}$, has the single-sentence DLS property, and is Barwise compact?
I'd also be interested in what happens if we weaken the definition of Barwise compactness to allow replacing "admissible set" with "transitive model of $T$" for some first-order $T\subseteq \mathit{Th}(\mathsf{HC})$.
