Is the Klingon word for a bathroom that doesn't contain a toilet also puchpa'?
A bathtub is a DoQmIv'a' greater sink. More specifically, it's a Say'moHmeH DoQmIv'a' greater sink for cleaning. A room whose purpose is bathing in a tub might be described with Say'moHmeH DoQmIv'a' pa' room of the greater sink for cleaning. I don't know of any "official" Klingon word for it.
A shower is a bIQ ghaywI' pa' water sprayer room. Many puchpa' also have a bIQ ghaywI' pa' inside them. This one is an "official" Klingon word.
Would a general room for cleaning oneself (regardless of method) be a Say'moHmeH pa'? And if so, is it an official word? (I can only dream of having a wet room...)
Say'moHmeH pa' means room for cleaning. It's not an "official" word in that we have not been told that Klingons consider it a lexical item that would appear in a dictionary.