First, please forgive my grammar, english is not my mother tongue so I might make a few mistakes...
I figured I might needed to go to the Saginaw Mental Institution to cure my Blue Rot because I saw somewhere that the weird blue cult-ish people were involved and it seemed like their headquarters. I decided to sneak in and disguise myself as one of them, talked to Liza and then went in the auditorium to accuse the Baron. I succeeded but I was quite sad to see King Elias die :( From what the text after I survive the trial (i got the Water one) says, it seems that Liza was lying to me and that she is involved in King Elias's death. He seemed like a pretty alright chap so I want to avoid having his death on my conscience. I went on the wiki and the way it was written, it seemed like the only way to talk to Elias and therefore not having him die is by getting caught by the guards at the entrance.
So my question is : how to avoid Elias's death? Is there any other way than getting caught by the guards, like accusing Liza instead of the Baron?
Saginaw is still one of the most complicated encounters as far as I'm concerned, and I've played it many many times. So I'm never going to claim being 100% sure on my answers when it comes to it. ;)
That said, I think you need profess your faith at the entrance in order to get the ending you want. Then, after the trials, you can keep Elias alive either by not answering him or by telling him the church is doomed.
There are so many different sides to this coin though, Saginaw can go so many different ways, and there many different little tidbits of information you can get from there (about Philip's past, about supernatural stuff, about the Frogs, etc). So the more of them you try out, the more you will get to learn really. :)