While electronic health records present a rich and promising data source for observational research, they are highly susceptible to missing data. For settings like these, Seaman et al. (Biometrics 68 (2012) 129-137) proposed a strategy wherein one handles missingness in some variables using inverse-probability weighting and others using multiple imputation. Seaman et al. (Biometrics 68 (2012) 129-137) show that Rubin's variance estimator for averaging results across datasets is asymptotically valid when the analysis and imputation models are correctly specified and the weights are either known or correctly specified. Modeled after the approach of Robins and Wang (Biometrika 87 (2000) 113-124), we propose a method for asymptotically valid inference that is robust to violation of these conditions. Following a simulation study in which we demonstrate that a proposed variance estimator can reduce bias due to model misspecification, we illustrate this approach in an electronic health records-based study investigating whether differences in long-term weight loss between bariatric surgery techniques are associated with chronic kidney disease at baseline. We observe that the weight loss advantage after five years of Roux-en-Y gastric bypass surgery, compared to vertical sleeve gastrectomy, is less pronounced among patients with chronic kidney disease at baseline compared to those without.