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If I understand this question correctly. You are basically estimating a precision matrix that varies over time; and you would like to draw a comparison between multiple estimated precision matrices. 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. Depending on what kinds of differences you are interested in, are you interested in an overall difference or differences in specific rows or recovering the exact support of the difference ? If the first, you might be able to use existing literature on testing covariance matrices. 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.