Convert a binary neural network classifier to one verses all classifier
You may keep N neurons and Sigmoid and treat is like multi-label though your data will always be multi-class. So it should learn like multi-class

Create your own One-vs-One

Have 3 copy of the same model
Fit with A|B, B|C and C|A respectively
For new data, you need the average of two for each Class probability


Use scikit-learn Classifier

OneVsOneClassifier Link
OneVsRestClassifier Link