What's the most effective way to measure the accuracy of my multi-class classification NN?
Keras has some accuracy metrics for multiclass neural networks.

CategoricalAccuracy
Top K  Categorical Accuracy
Sparse Top K Categorical Accuracy

There are some others if you want to implemented from scratch. Taken from: https://www.sciencedirect.com/science/article/abs/pii/S0306457309000259

These work for most problems rather nicely combined with a good cross validation approach and leak prevention.
Having an imbalanced dataset always makes it challenging but picking a metric that doesn't punish results too much like Top K Categorical Accuracy combined with n-fold cross validation like this article suggests, is often enough to have a solid evaluation procedure that you can work on.