After you've trained the model as a dlnetwork, there is nothing stopping you from converting it back to a traditional SeriesNetwork and adding a classification layer for the purposes of prediction.
You could also define for yourself a custom layer,
to replace the traditional output layer. The idea is that you replace it with a layer which stores your class names and which does essentially the same thing as the legacy classification output layer, but because it is derived from the nnet.layer.Layer parent class, it would be compatible with dlnetwork(). With this approach, you would write the forward() method for the custom layer to ignore the class names for the purposes of training, while the predict() method would be written to use them.
