Using SPI without driving MISO
In microcontrollers, the GPIO pins have a multiplexer to select what the pin is used for.  When you use the Arduino SPI library, the initialization function will set the multiplexer to MOSI for that pin.  You should be able to run your own initialization for that pin and overwrite whatever the SPI initialization function did.  Just make sure to call your initialization after SPI.begin().  This is the simplest solution.

If that doesn't work for some reason then you can dig into the source code for the Arduino SPI library.  This isn't too hard either.

Bitbanging is the hardest solution and shouldn't be necessary for your application.