Definitely not fat. Out of shape and I think underweight actually. Like you say, maybe you think she's fat because of the fur, but her neck is thin, her back looks like it may be peaked and her rump is definitely not roundish. Also, when a horse is out of shape, it can be easier to think they are fat because things are nice and tight and shiny.
BTW, it's hard to tell with the photos, but maybe her front toes are long?
As for stall rest, unless it's a broken bone (generally speaking) I'm completely against it. I've seen the harm it's done and I say: turn her out with free choice hay! It'll do her far more good than being cooped up in a stall.