What's more appropriate and faster for outlier detection? Classification or Regression?


I’m working on a data set where I only care about “outliers” from the norm. The data is comprised of long time series with gaussian distribution, while once in a while there is a large outlier. I would like to classify this into: “normal”, “large positive outlier” and “large negative outlier”.

As far as I recognize this problem that can be solved either:
(1) via regression, using MSELoss, and classifying the prediction “manually” at the end into one of the three classes. The target in this case would be the outlier value.
(2) via classification in 3 classes, using CrossEntropyLoss. The target in this case would be the class number directly.

What do you think is more appropriate here? Which one is faster?
Both seem to converge to just predicting “normal”, because that’s correct in 99% of the cases. Which model should I use and how can I force the model to make predictions on outlier cases as well?