Is there any solution to run this display via a 2/3 wire connection with 2 of these 595?
Yes, I am sure that you can do this if you really want to. It will be fairly slow and complicated.
Buying a ST7920 style display is a lot simpler. It can do everything that a KS0108 display can do. And it can even work like a regular HD44780 character display.
It runs on native SPI . Most Ebay modules are 5V operation.