Try this: use the TPIC6B595 OE to disable the outputs when switching.
Use direct port manipulation to make it go faster.
So that OE is bit 2 and SS/Latch is bit 3, both on port D:
PORTD = PORTD | B00000010; // bring OE/ high
// xfer data
SPI.transfer (B00000000);
SPI.transfer (B00000001); //leftmost, frontmost corner cathode column
SPI.transfer (B00000001); //bottom anode plane
PORTD = PORTD & B11111011; // bring SS/latch low
PORTD = PORDT | B00000110; // bring OE/ an SS/latch high