Is it feasible to train a single model on 150+ classes?
A Classifier's quality depends from the number of available observations for each class. If you have 300 images for each class, as you said above, it might be enough (it depends from the quality of your data).
In any case, it would require a massive amount of data augmentation.

Training a Network on 240 classes could be very challenging, if you don't have a proper infrastructure. That all depends on training times, but I'd suggest you to break down the problem in smaller subproblems. That would also make debugging and improvements easier to implement.