convert predict_proba results using class_weight in training
The correction that you are talking about is called probability calibration -- you want the "real" probability of an observation being in each class, right?

The two most common approaches to probability calibration are Platt Scaling and Isotonic regression. Since you are training on a balanced training set (which is the right thing to do in your case because the original dataset is unbalanced), you can apply these techniques afterwards on your test set.

Sorry I can't explain the techniques fully here, but hopefully knowing the names of these terms gives you a starting point.