Let me start with an important point; what other criteria than classification success can U use to identify my cluster separation performance?:
- Classification as an indicator for clustering performance has an internal paradox. If you have the classification the clustering question does not apply anymore. These two concept are coming form two totally different philosophies so I would say be careful if you have already understood the concepts (what you say may make sense in semi-supervised learning which is not in your tags so I assume there is a misunderstanding).
- Clustering has no performance evaluation! this is the problem I see most of data scientists struggling with. In practice you may define a good criterion (but honestly, what is good?!!) and deliver your solution, but in research schema there is no evaluation for clustering as the question itself is not well-defined i.e. you never have label to be sure who is who so you need to define closeness of points from which the problem starts; how close is called closeness?!!
- Be careful about Curse of Dimensionality while using k-means for time-series clustering (I'm not sure how you do it).
After these points let's have a look at your question.
What are time-series? If time-series are pretty non-stationary or simply speaking is the dynamic behind variation is complicated enough, then there is not a one-to-one map for time-series characteristics and time points (imagine a simple ECG signal. It's pretty simple but if you explore research community you will find super sophisticated methods for feature extraction on ECGs. I'm pretty confident finding a time point at which ECGs differ is almost impossible.). You may extract features from your time series or embed it into some n-dimensional manifolds and look at it. In best case you may find some time-related features which describe your time-series and you may find some time-related criteria at which time-series differ (however I'd say it's not likely to find them).
Assuming time-series are pretty well-behaved (!!) with a simple dynamic (should be super simple). Then a solution might be to define a distance function of time-series which outputs the pair-wise distances of all time-series as a single score. Then the maximum of this function returns the time-point at which these time-series are pretty distinguishable.
If you provide more information on your data I might be able to give a more detailed precise answer.