Use unused SPI MISO pin for I/O on ATTiny
The SPI MISO function is nothing but a digital input pin.

The question does not specify whether the pseudo-SPI LED peripheral provides a handshake or other data on the MISO line at all. Assuming there is some initial data at set-up if at all, but no further use of the line by the peripheral during normal operation, the following should work:

Connect one of your 3 data source lines, OR'ed with the MOSI coming from the SPI peripheral, into pin 6 of the ATTiny.

During SPI set-up, ignore (or do what's needed with) the signal coming into pin 6. After that, simply read Pin 6 as you would any normal digital input pin.

If the LED peripheral works fine without connecting it to the MISO on the ATTiny at all, then you can skip the OR connection, and simply treat pin 6 as a standard digital input.