Multi-label classification based on single-label dataset
I would train it using the loss appropriate for multi-label (sigmoid activation/binary cross-entropy loss instead of softmax activation/categorial cross-entropy loss).
The model will give you probabilities per label, so it's up to you to decide how to interpret them in your application.

Having said that, some multi-label problems have mostly single labels, much fewer double labels, and almost no labels beyond that. For problems like this, you could try a single-label model and see if it works better than a multi-label one. Or have an ensemble :)