It is possible to launch Cortana from UWP using the Launcher.LaunchUriAsync() method, using the invocation URIs, as detailed here .
Cortana also has a full screen mode which is activated when there in user input for 10+ seconds. In that case, "Hey Cortana" opens cortana in full screen on the primary display. Is there any Cortana invocation URIs to start Cortana from UWP in full screen always?
Cortana does support a few "deep linking" capabilities. You need a registered protocol handler installed (that is the case for W10 desktop by default.)
However controlling screen mode isn't one of them.
Feel free to add a suggestion for this via feedback hub.