The LCD image does not have a standard desktop on it. It runs a special script for the LCD. This is available to the hdmi port but it is only expecting touchsceen inputs. While you can add a desktop, It won’t display the standard desktop on a monitor at the same time as the LCD GUI without some complex changes, otherwise it is one or the other. You could enable VNC and remote to it and use a mouse. I have tried VNC to the LCD image but did not not test a virtual mouse.
The standard image fully supports the desktop.
See here:
VNC to LCD image