There is no "Havent got" in propper English. While constructing a negative sentences in Present Perfect, you always use past participle.
Present Simple: Usually I don't have much time.
Present Perfect: I haven't gotten much time recently.
In order to develop more understanding, about rules of any language, you should accept the fact, that being a native speaker of any language, doesn't mean that you use it perfectly well. On the other hand people shouldn't be submissive to a language - it should be, the other way around.