Low cost could be a PSoC4 which has built in a proximity sensor, but I do not know if that will work as expected in a bathroom. Get hands on a Pioneer kit www.cypress.com/ to build a proof-of-concept. Later on you may use a Prorotype Kit-049-4xx to have prototype to start with.
Additionally there is a brand new chip with a Kit here secure.cypress.com/, but I do not have any experiences with that.
Standard solutions as IR reflex or light barrier could easily performed with a PSoC4.
Here is a basic approach done in PSOC 3, should be able to try
out on PSOC 4, substituting SAR for DelSig. Additionally PSOC 4
has a fast multiplier should you need to do some basic DSP.