Given that both devices have a slave SPI interface, you can't. (At least I've never run into any SPI RAM or SPI LCD that could act as a master.)
The usual way would be to set up DMA transfers within your MCU, which would take almost no CPU time (apart from starting the transfers, a couple of cycles if done right). If it doesn't support DMA, you're out of luck. Of course you could always add some kind of small FPGA/CPLD in between to take care of the transfers, but I guess that would be way beyond what you intended to do.