The possession of an aspiring champion by a greater daemon is entirely different from the process of ascending to princedom. See Lucius the Eternal for an example of how much control the champion has left after the greater daemon shows up. And on top of that, the summoned greater daemon doesn't care a bit about preserving the champion's body.
Which daemon prince model are you asking about? Do you mean the most recent metal CSM daemon prince, the new plastic one, or one of the old and scrawny metal daemon princes?
The list of reasonable things to do:
1. Rather than use a current do is take one of your winged Greater Daemon models and field it as your generic summoned Greater Daemon. If you'd like to be helpful, remind your opponent that even though the model has wings, the version in the codex is too weakened to use them for anything. The key is that you can look at GW's website and note that all four types of greater daemons are marketed as "summoned" versions.
2. Convert one of the new plastic daemon prince kits into something generically greater daemony. The key is converting the model.
3. Take one of the old and scrawny metal daemon princes (not the giant-marine-in-powered armor princes, one of the old Belakor, Azazel or similarly old models) and convert slightly into something generically greater daemony. The key is converting the model.
Things that are not good:
1. Placing an unmodified Daemon Prince model on the table and calling it a summoned greater daemon. This is not good because you're taking a model which looks like one thing (a daemon prince) and calling it something else (a greater daemon), and that's just going to cause confusion. The key problem is that you haven't done anything to the model to indicate that it's supposed to be anything than what it by default should be: a daemon prince.