Input for LSTM for financial time series directional prediction
I think there's a partial answer to your question, about the part: "the sequences it's fed are all parts of the same long sequence"

You can train your LSTM model in stateful mode. There's a clear explanation here: http://philipperemy.github.io/keras-stateful-lstm/

In short, in stateful mode, the model will remember what it is previously fed and will also use that information for training. That way, it can sense that the current input is a part of a long sequence.