UPDATE Dec 2019: Based on Cinelli & Pearls response to the OP (& associated paper), it does indeed look like all the relevant information can be integrated into a DAG model.
Over the course of this thread, I came to the impression that an unnecessary focus on identifiability was the main root problem with the OP. Now it looks like that was probably wrong. However, based on the Cinelli & Pearl paper, it does look like causal DAGs + Bayesian probability (or even non-Bayesian probability, for the example at hand) are all we need for this use-case.
UPDATE Dec 2019: Based on Cinelli & Pearls response to the OP (& associated paper), it does indeed look like all the relevant information can be integrated into a DAG model.
Over the course of this thread, I came to the impression that an unnecessary focus on identifiability was the main root problem with the OP. Now it looks like that was probably wrong. However, based on the Cinelli & Pearl paper, it does look like causal DAGs + Bayesian probability (or even non-Bayesian probability, for the example at hand) are all we need for this use-case.