They also have no reason NOT to swap his name. Why not? If it's easy they might as well. We will see, it could go both ways.
What I was saying is that apart from civilians and fodder, all the influential people will know, maybe even random Celestials. If someone who knows he's dead will see him alive they won't give him any particular clearance, because he's lost his rank.
What they will say about Saturn to public is,
A- if they will say nothing about his death and try to hide it or that he is just disappared then all rights will be available once they see him alive.
B- If they say something like pirates killed Saturn, then its same as they will treat him same once they see him survived and it was a wrong info.
C- If they say Saturn failed and he is punished for his failure, (which is low chance since that would reveal there is someone above 5 Elders, since no one could punish a 5 elder), they would say that to public then there might be a small chance they might remove his rights despite being dead, they might still not bother with it since he is dead.
If random Celestials don't know about Imu, then they need to make up something different about Saturn how he died.
Obviously they also can't tell Marines about Imu's existence.