I believe the SPI hardware configures it as an input automatically so it won't be usable as an output - but I could be wrong and it can be overridden with pinMode() - try it and see for a definitive answer! (Reading the datasheet might produce an answer too, but it might be easier to try it).
Alternatively software bit-banged SPI would do the job.
Isn't the SPI a hardware function?
See Section 18 of the data sheet:
"When the SPI is enabled, the data direction of the MOSI, MISO, SCK, and SS pins is overridden
according to Table 18-1 on page 168."
not "typically" - that is how the hardware is built in the ATMega.
So you enable SPI by writing the SPI enable register, and the configuration of the pins is determined from there.
I meant "typically" in terms of the use of MISO and MOSI
I accept that the hardware also uses them that way
I suspect that the OP isn't aware that even the master has to listen sometimes