I *think* that if you ground SCLK, then the SPI port will just sit there waiting for the next clock cyle and data. It doesn't enter some kind of special shutdown mode because it's not like a UART where both sides know what the data rate is supposed to be. It just behaves as if the clock signal was in the LO part of a periodic clock train.
Because you want to communicate with other devices you will need a separate special mechanism to pull the SCLK of the LCD low, and a way to disconnect it from the SCLK in the other devices so that only the LCD SCLK will be low. Like maybe a NMOS to pull the LCD SCLK low and a series resistor to isolate the LCD SCLK so when the transistor pulls low it does not pull the entire SCLK line low for other devices.
This is just my opinion, I've never used LCDs before.