In Present tenses^x, American English^x, Negatives^x

The idiom is "have got" in both American and British English. It is a purely idiomatic alternate for "have" and is not much used except in the present tense. That is, "have got" is the present tense of this idiom, even though "get" is the present tense form of "to get", and the past is "got". ...

Yes, you are right - this is a matter of British and American English: In American English, the forms usually are: get (infinitive) got (past tense) gotten (past participle) while in British they are: get (infinitive) got (past tense) got (past participle) The "inappropriate" use you are...