Multivariate time series prediction with binary target
I recently trained some multivariate LSTM regressions on a relatively slow, CPU-only laptop, and it worked. Training time didn't take too much, it's definitely feasible.

About your problem, I would definitely use some RNN architecture. LSTM are the most powerful RNN cells up to now, they have more memory and are able to learn longer sequences than GRUs, but they are slower than them. Choose based on you preference/circumstances.

The final layers must be Dense(), and the final output layer should have two nodes with softmax activation, to perform the binary classification. The most appropriate loss would then be some crossentropy measure.