There are some methods out there for two sample testing of covariance matrices but no one has specifically looked at testing for conditional covariance matrices. Are you interested in an overall differences between covariances or differences in specific rows of the covariance matrices or recovering the exact support of the difference ? If the first, you might be able to use existing literature on two sample testing of covariance matrices. This would definitely work for point 2 in your question. This is really an open area of research and quite under explored in the time-varying setting. An alternative to KL-distances is using the maximum t-statistic across all the entries of your conditional covariance matrix.