I'm going to let wiser heads answer your overflow valve question - sure seems like a darned good idea to me! But I've filled mine up, and it DOES back into the shower and until the shower is overflowing it wouldn't get on the floor. Luckily I was standing IN the shower when it started burping and backing up so I rinsed quickly, got out and emptied the gray tank.
The fact that yours flooded your floor without filling up your shower pan leads me to believe you might have a cracked drain or poorly attached drain in your shower... it may be running out right before it should be backing up into the shower. There's probably a rubber gasket between the pipe and the shower itself. The best way to test that would be to put the plug in the shower drain and fill the shower. If the water level goes down it's the seal between the pipe and the shower's floor drain hole.
I don't know whether you're going to have to go in through the belly pan though I suspect you will, or whether you could see the problem by opening the panel under the closet (in the 25 FB it's on hinges). It's a door just like the closet door - but no handle. I use a knife or scraper to pop it open.
Contrary to common practice I often DO leave my gray open - I fulltime and shower til I run out of hot water. I close it once a week when I'm doing black tank draining. Before I leave a campground I close the gray and turn the sink on full blast so there will be a lot of whoosh to get the coffee grounds out of the bottom of the gray.
Sympathy Karma to you.
Today is a gift, that's why they call it the present.