How to break down large SVM classification model?
I think that using SVMs for your model is the main problem. 

SVMs (linear or otherwise) are originally intended for binary classification. There are various procedures for extending them to a multi-class problem. The most common methods involve transforming the problem into a set of binary classification problems. The most common strategy is using "One-Versus-All" classifiers (OVA classification), and choosing the class which classifies the test data with the greatest margin. Another strategy is to build a set of One-Versus-One classifiers, and to choose the class that is selected by the most classifiers. Both these strategies are computationally expensive both in training and in testing.
I know there are other, more complicated methods, but I don't think that any of them is intended for such a high number of classes.

The most trivial way would be to use a neural-network model for such a problem. If you want to use one of the more classic algorithms, perhaps using decision trees will work (or even k-nearest neighbors).