Hi,
More than one way to do this...
See an application note for details:
and "Cascade Operation of Multiple TPIC6595s"
Basically you can shift data into multiple chips, and after they are all loaded with the right data you can enable the outputs.
At least you could set up sets of 3 .
Somewhere I saw little breakout boards for HC595 chips; maybe you could solder the high-power version on those to make it easier? Like THESE:
Arduino Library here: GitHub - infomaniac50/Arduino-TPIC6B595
Let us know where you go...