From what I can find right now out would be "et"- (which is a prefix), and house is "coa". I guess you could put them together and get "etcoa" for "outhouse".
I am getting these from a Quenya wordlist I found on google. It may be different in Sindarin.
As far as bathroom goes, I think theirs would be more like wind chambers, if you know what I mean ;^)
Thus we could have wind=gwaa or gust of wind=gwam combined with chamber=sam or samm to have gwaaasam or gwamsamm....
Get thee to the Gwamsamm, Elrond!
Hmmmmm....this is going to be fun!