 Solution

Treat the extra measurements as auxiliary variables for multiple imputation. Include alcohol consumption and disease history as predictors in both the $C$ and $P$ conditional distributions, together with age, sex and the other incomplete response. If these auxiliary variables themselves have missing entries, give them appropriate models and update them within the same multiple imputation by chained equations sweeps. Preserve observed entries throughout.

Useful auxiliary measurements can improve predictions, precision and the plausibility of missing at random. Their association with the missing responses reduces uncertainty in the imputed values; association with nonresponse may explain some selection that otherwise depends on unobserved values. This can reduce selection bias under the enlarged missing at random model and reduce between-imputation variance. It does not guarantee elimination of missing not at random dependence. They need not be added to the final substantive logistic regression merely because they help imputation: the final analysis should still estimate its intended association.