Δ-Interpolation doesn't imply Beth property in abstract model theory: a simple counterexample?

Question

It's known that for an abstract logic $\mathcal{L}$ (for definitions cfr. Barwise J. & Feferman S. (eds.), Model-Theoretic Logics, Springer-Verlag, 1985) having Δ-Interpolation (also called Suslin-Kleene property) doesn't imply having Beth property. The only counterexample I've found is the logic $\Delta(\mathcal{L}_Q)$, the $\Delta$-closure of $\mathcal{L}_Q$ (first-order logic plus the quantifier 'there exist uncountably many'), which by definition of $\Delta$-closure operation has $\Delta$-interpolation, but it doesn't have Beth. The problem is that the only proof I know of is from Shelah's The theorems of Beth and Craig in abstract model theory, III: $\Delta$-logics and infinitary logics (Israel Journal of Mathematics 69/2, 1990), but it's a very general and complex one and the counterexample I'm interested in is only a simple corollary of it. Does a simpler and more specific/direct proof producing a counterexample only to the implication above exist?