Strong overfitting accompanying strong class imbalance
With complex data it's rare not to have any overfitting (or underfitting). Ideally one wants to avoid strong overfitting, and if given a choice between two models it's clearly safer to use one which isn't overfit. But if it's impossible to avoid, from a practical perspective there's no reason not to use a model for this reason, in my opinion. The problem with an overfit model is simply that it's going to perform poorly on fresh instances, so the worst that can happen is the performance that you observe on the test set.
Just an idea: I don't know if it makes sense with your data but it might be worth trying one-class classification on the minority class. This could be a way to avoid the imbalance issue, and could be followed by a second stage of regular binary classification.