This is a good question, I would be interested in other answers.
Random forests are more interpretable than may first appear see Obtaining knowledge from a random forest. As an example and as mentioned in one of the answers from that question, the random forest you can get the importance of variables, say using the importance function of the R randomForest package.
Conversely, using a single logistic regression model may seem more interpretable than it actually is since unless there really are those exact regressors generating the output (unlikely) then probably a few slightly different models will be acceptable or plausible. Further, changing one regressor will often effect other regressors in the model if they are present, and how much to change a regressor (one unit, one percent?) may not even make sense for a given regressor, with a further complication being the variability of regressors is unlikely to be the same.
I would fit both a logistic regression and a random forest and look for similarities and differences in their results, augmenting this with any other domain knowledge or plots that help elucidate the data.