It sounds a bit like "the pen of my aunt is on the table" so beloved in old French lesson books. I think the bath tap is dirty is a better translation.
It's more like a formal construction but it's correct. "The bath tap is dirty" does not imply that it's the bathtub's tap (faucet). "The bath tap is dirty" sounds more like "The sink's faucet is dirty". Native speaker.
The bath tap, the bathtub tap or the tap on the bath are preferable to the "Tap of the bath"
singular cadă / cada
plural căzi / căzile
singular căzi / căzii
plural căzi / căzilor