My 'simple' solution is turning out to be incredibly frustrating.
I have no idea why but the DEAD stat is mysteriously applying to characters.
I do not have any idea about this.

But I have an idea how to do it without the mark "Dead".
Maybe we should realize a system "Purgatory"? After the death a character is transferred to some location. Character is automatically started trigger on this location. This trigger opens a menu that blocks any control + our character use a dead- animation? What do you think?
Developing this idea we can make a menu that intercepts the camera and shows a grave, or monument, or may be a skeleton. Or just a big inscription "You defeated"
Yes, by using the console, you can return the character, but it's a cheat, and, in principle, that's fine.
I have this setup working. It's very minimal and basic right now so I'm going to polish it up a bit before I send it your way.
Basically, what it does right now is one of two things:
1) If the player has a Vitalyser it respawns them at the beginning of the dungeon. I scrapped the Vanilla UI entirely and don't know how to respawn in place without it. But the Vanilla UI was causing problems I didn't understand.
2) If the player has no Vitalysers, they respawn in the QA Test room. We can always change the location later.
I have this setup working.
Hooray!

Thanks gytfunke. I think everything will be great.
Does that mean it works?
Yup, it works! Need some work on UI and accelerate the resurrection, but overall everything great.
This is a cool mod. I've always been on the lookout for something in-between normal mode and hard-core. Having a few lives is a great option. Also brings up the opportunity for some cool risk / reward scenarios.
I think it may be easier to permanently transform the character into the grave after death? I can create some unit - the Grave which can`t move.
Oh, gytfunke, is there a way to only 1 items are removed from the inventory? Now, after the resurrection removes all resurrect items.
I think maybe add a function - "Run Vitalizer". Then you get the quest item - "Working Vitalizer", which will be removed from the inventory.