What can I wash in the bath? This verb needs a direct object in English. Or we could use a phrasal verb such as "I can wash up in the bathroom." It's also more likely we'd say "bathroom," since *bath" usually indicates the warm water already drawn and ready for someone to bathe in.
Now, regarding the Latin. Shouldn't it also be reflexive? Something like, "In balneo me lavare possum?" Or are we trying to say that we can wash things in the bath (bathroom/bathtub)?
I think the intention of this sentence is to sum up all the things we can theoretically wash (that were named before - a dog, a friend, yourself etc.); so, yes, it sounds odd, but is technically not wrong.