For the augmentation i was only talking about this:

For a lot of stamps I have real camera photos, 10-20

If you are really going to have a large scale classification problem it is going to be quite similar to the large scale face recognition proposed solutions.
E.g. Glint360K dataset has 360k identities so quite similar to your 350k+ but with other tricks you can scale also to 10M 20M 30M 100M. See:

  
      

      arXiv.org
  

  
    

Partial FC: Training 10 Million Identities on a Single Machine 9

  Face recognition has been an active and vital topic among computer vision
community for a long time. Previous researches mainly focus on loss functions
used for facial feature extraction network, among which the improvements of
softmax-based loss...


  

  
    
    
  

  



  
      

      arXiv.org
  

  
    

An Efficient Training Approach for Very Large Scale Face Recognition 4

  Face recognition has achieved significant progress in deep-learning era due
to the ultra-large-scale and well-labeled datasets.
  However, training on ultra-large-scale datasets is time-consuming and takes
up a lot of hardware resource.
  Therefore,...