One new algorithm that you might not be aware of is Gloria. It is not neural network based as your current approach, but is state-of-the-art in a sense that it significantly improves on the well-known Prophet.

Online traning is not yet available (i.e. updating existing models based on the latest new data point), but including a warm-start is on our roadmap for the upcoming minor release (see issue #57), which should speed up re-training your models with new data significantly.

As Gloria outputs lower and upper confidence intervals simple distance-based anomaly detection is very straight forward. Based on the data-type you are using, you have a number of different distribution models available (non-negative models, models with upper bounds, count data,...). These will give you very reliable bounds for precise anomaly detection. With a little bit of extra work, you will even be able to assign a p-value like probability to your data points of being an anomaly.