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We present an exactly solvable effective model of a double quantum dot coupled to superconducting leads. This model is a generalization of the well-known superconducting atomic limit approximation of the paradigmatic superconducting impurity Anderson model. However, in contrast to the standard atomic limit and other effective models, it gives quantitatively correct predictions for the quantum phase transition boundaries, subgap bound states as well as Josephson supercurrent in a broad range of parameters including experimentally relevant regimes. The model allows fast and reliable parameter scans important for the preparation and analysis of experiments which are otherwise inaccessible by more precise but computational heavy methods such as quantum Monte Carlo or the numerical renormalization group. The scans also allowed us to identify and investigate new previously unnoticed phase diagram regimes. We provide a thorough analysis of the strengths and limitations of the effective model and benchmark its predictions against numerical renormalization group results.