If I was in your shoes, it seems like of those eligible, you could run a propensity score model to get weights to use in your outcomes model. As you mentioned, you may not have access to all the variables but this would be better than a naive or conditional model. For the propensity score model - gradient boosted trees or ensembles are best - but logistic reg would be sufficient. For the outcome model you can just use a simple regression with propensity scores as weights.

Once you get the weights, many like to use the standardized inverse weights. So use inverse of (probability and times prevalence of which group they are in).