"Someone spilled a few drops of fake blood on the LCD screen and now there's a dark stain that is over roughly half of the monitor."
Did the liquid get to the edge of the LCD and possibly get under the screen, or is it strictly on the front surface?
If it got underneath you are probably hosed.
Fake blood is usually corn syrup and food coloring. Shouldn't be anything that could 'get into' the plastic, just a really stubborn food stain.
I think your going to have to get creative.
You might try eyeglass cleaner, most is plastic safe. After traditional cleaning, maybe lay a patch of paper towel dampened (lightly) with eyeglass cleaner on the screen and let it sit for a while. See if it will draw the coloring up. If you get any color on the paper towel, keep doing this with fresh towels. it might take a while.
If this doesn't work, you'll have to ramp up the strength of the cleaning agent.
You might want to go to the dollar store and by some cheap calculators or something with a similar LCD screen to try out cleaners on before trying them on the camera.
There are a ton of homemaker tips web sites about cleaning, stain removal. You might want to check them out for ideas, since you're basically dealing with a food stain.
I have a system, it has stuff in it, and stuff hooked to it. I have a camera, it can record stuff. I read the manuals, and know how to use this stuff and lots of other stuff too.
You should be suitably impressed...