Let's analyse the two versions separately.
My father said that I have to clean the bathroom, but I don't want.
The problem with this sentence is that want is a verb of incomplete predication in this sentence and such verbs demand a subject complement or an object complement to make a complete sentence. Since such isn't the case with this sentence, it is grammatically incorrect.
My father said that I have to clean the bathroom, but I don't want to [clean the bathroom.]
In this sentence want is a transitive verb, and because transitive verbs demand an object, the above sentence is a grammatically correct sentence. The object of the sentence is the implied infinitive phrase to clean the bathroom (It's not incorrect to leave out part of a sentence like this; this is called ellipsis in grammar.)
Therefore, this version of the sentence is the correct one.