As Jindřich has said, Q, K, V come from previous computations, they are not trained directly with backpropagation. However, the weights $W_i^Q, W_i^K, W_i^V$ are trained directly with backpropagation.
Expanding on this, in the "Attention is all you need paper", in the self attention used by the encoder and decoder, Q, K, V are the same matrix.
If I got it correctly, you might have the following scenearios:
labeled samples with tag A or B per date-time index, having as informative attributes the present + some lag values of interest --> (this would be a standard classification approach without time ordering bein necessary)
sliding window of samples (what you mean by subsequences) --> here ...
In principle, it is possible to reuse the special tokens as you describe.
However, according to research, you should not freeze BERT, but fine-tune the whole model with your data, in order to obtain better translation quality.
Another option would be to reuse just the embeddings instead of the whole model.