Movie Night Questline bugged in New Game Plus

As far as I can tell, while you can sell Liam's armor, the cybernetic for Alain, and the humane trap, you cannot sell or destroy any items that fall into the special category such as the ships, and the movie night quest items.

I did find that, if I pushed through things and "completed" Liam's final request for the movie then it would count as finished.  I could talk to the others and it would play their scene with me given the item (even if I never got the quest from them, like Lexi and Jaal) but it didn't seem to effect anything.  I didn't actually do the movie night scene after doing that, however.

So it may be safe to speak to Lexi and skip immediately to finishing Liam's final request, since you'll already have the item in your inventory.  Might give that a try and see what happens.