How to find "regions" with high purity
A way to explore your dataset would be to use different techniques and see how they preform. The first approach I would consider is a tree based learner of modest complexity (or at least not boosted), like C4.5. It's possible (or perhaps likely) that won't perform as well as some more powerful approaches, but it has the added benefit of being relatively readable.

If you are familiar with a decision tree visualisation, you'll know that you can identify (leaf)nodes with high(er) impurity, and a large(r) population quite quickly. You can find an example of such a visualisation here. Those nodes can help you understand your domain, and possible how to improve performance (For instance by boosting, something LGBM already does for you). 

If you want to optimise the LGBM straight away, another thing that could help here is to cluster your dataset (kMeans could be a good starting point) and then calculate performance metrics per cluster. Possibly you could also consider clustering the scored items, including the classification, or even the evaluation (TP, FP, TN, FN). If used right that could help identify clusters of false negatives for instance, and the commonality they share. Disclaimer: I have never done this myself.

PS: Lot's of times I find it helpful to use KNN (over a subsample of the feature space) and Naive Bayes. They can quickly tell you something about the global correlations. Other golden oldies are feature importance rankers and a correlation plot.