Is it possible to further compress a Base64 PNG String?

The simple answer: No - not without loosing the "printable string" nature

Usually PNG already uses sophisticated compression like it is used in ZIP files. Therefore compressing it before applying the base64 encoding will give you only very limited size reduction.

Applying the compression after the base64 encoding will make it to binary data again - in this case you could just skip the base64 encoding step.