Keeping her alive is pretty easily explained using just simple logic. Rayet used a necklace to choke out the princess, a necklace which she immediately let go of once the princess lost consciousness. The main reason choking causes death is because they end up severing the spinal cord, but a necklace isn't going to do that. Right now the princess is either unconscious, now breathing because the necklace was let go, or worst case scenario, her heart stopped which gives her 6 minutes before brain damage kicks in. More than enough time for someone to realize something happened to the princess (already happened since the damned ship fell out of the sky), Eddelrittuo to run to the shower with Inaho following her and Inaho (or really any character) performing CRP while the doctor is called to help.
Of course, that's only if they want to follow along what should happen. If they really want her dead, they will ignore almost all of that.
I personally think they'll keep her alive, mainly because they put a big focus on the Princess seeing and knowing it was Rayet choking her. They wouldn't have done that unless It was going to come up later as a conflict item between Rayet vs everyone after the princess tells.