Looking for binary class datasets with high class imbalance, that also have intra-class imbalance in the minority class
The imblearn.datasets package (documentation is here) has a function called fetch_datasets() which is described as:

fetch_datasets allows to fetch 27 datasets which are imbalanced and binarized

I do not know them in sufficient depth to know whether they meet all four of the criteria you've listed, but these may be a good first place to start with.