Observational data with diagnosis partially determining treatment (imperfect instrumental variable?)

Here’s the setting - we have a retrospective dataset of patients undergoing a certain surgery. The surgery has two variants, let’s call them A and B. We are interested in comparing a set of outcomes like length of hospital stay and blood loss in each variant. The choice of variant is based on clinical judgement and thus there are inevitable selection effects biasing the comparison. If that was all, we’d have little options beyond just reporting the results descriptively. We however have an extr...