Does Arduino SPI use up a timer
The SPI module has its own baud rate generator so it does not consume any of the timers.  And the baud rate generator is only used in SPI master mode.  In slave mode, it is synchronized to the clock of the external master.  The only issue you may have with SPI is dealing with the data byte in a timely fasion as it is not possible for a slave to delay SPI like it is with I2C.