A bathroom is usually a modest space, very easily simulated to scale with some grid paper and scissors
Yeah, but then you don't get to ray-trace the reflections of the shower curtain on the porcelain toilet bowl or chrome accents. I mean, you can do some mock-phong shading with a few nice pencil crayons, but it's not the same thing.
If you need to browse the space in 3D and for "FREE" I'd probably also recommend Sketch-Up - it's pretty generic and can be used for anything. I've seen it demonstrated in person by their company employees a half dozen times and I've always been extremely impressed. Looked pretty straight forward for basic use, plus there are a ton of tutorials available on the net, including some really nice video ones.
Truly geeky would be to write your own app over the next couple of years while using a bucket in the meantime.