I would try calling a few times. In my (limited) experience, significant schedule changes do allow for a little more flexibility in changing your itinerary - I had to do the stopover/connection change myself. In this case you're going to need to either change routing or carrier - I'm guessing it might be marginally easier for them to keep you on CX with another routing, but I don't see why they couldn't switch to UL and keep the routing.
(As I write this, I realize that you might also have the issue that UL recently joined the alliance, post Explorer awards... not sure if that is what the agent was referencing.)