No this is not possible today.
One quick way to accomplish the scenario is to have the UWP write the command string to the local app data/settings, which is shared between the two processes. So the full-trust process can then pick up the command string after its been launched from there.
A more complex solution is to establish an app service connection between the two processes, and pass the command string via that connection. This will be helpful if you need to keep communicating back and forth between the two processes.
If for some reason you can't change your existing full-trust process code, you could add an extra EXE to your package that just reads the command string from the app data and then launches your actual full-trust EXE with those parameters (using Process.Start() or something equivalent).
EDIT
I have posted some more details and an example on my blog:
https://stefanwick.com/2018/04/06/uwp-with-desktop-extension-part-2/
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