However, it too would be fed from the Arduino via SPI and I absolutely do not know whether it is possible to connect two different SPI controllers to one Arduino.
You mean like an Ethernet card with an SD card on it? Both are SPI devices. Each uses a different slave select pin.
So, yes, it is quite easy to connect two SPI devices to one Arduino.
I imagine that it is NOT possible to connect both to the physical SPI pins.
Why not? As mentioned, you need different slave select pins, but the same SPI pins are used.
Unless the LCD is to be used for input, too, a Serial LCD would be a better choice, in my opinion.