Sorry I should have clarified, the TLC5940 library. The chip itself doesn't care how the signals come in.
Pin 10 is typically a chip select for SPI, which is a very loose protocol.
You could use a pin other than 10 with the TLC using the library's SPI mode, but you would have to modify the pinouts/ATmega_xx8.h file to change the "BLANK" pin to something else. The pins are defined by the library as hard coded bits to the ports on the ATmega328.
The tutorial you linked to has it's own way to working, maybe its best just to follow that.
Cant I just use the SCK and MOSI from the ICSP for the SCLK and SIN of the TLC? Is it a problem that the shield is also using it?
The other pins (10,9 and 5) are not being used by the ethernet shield so, no problem.
Did anything ever come of this? I'm trying to do the exact same thing, but can't sort out how to get the Leonardo Bit-Banging to the TLC5940. If you've figured out how to get that step working, I'd love to know