Migrainesbythedozen You cannot use the rules of other languages for British English. It does not work like that. I suggest you speak to a Professor of English Literature at your local university who can explain it to you. You are wrong, it's that simple.
I'm not "using rules of other languages" - please don't make things up to suit your narrative. I'm using standard grammatical rules relating to past and present tenses. Although its not as if they are uniformly taught in Britain, so I'm not sure why asking a professor of English literature would be relevant. Perhaps a professor of English grammar might like to answer questions from non-students a little more, although its equally unlikely. This is school level stuff, not university level. You must be able to distinguish between tenses in your own language in order to do so in other languages.
Ochon If you put some actual words after "I've gotten" I'm sure I'd get what you mean
"I've gotten a new hat" - meaning that I have already acquired a new hat. Its in the past.
"I've got a new hat" - meaning I've literally just acquired a new hat. Or at least its the present imperfect tense. You don't know exactly when the hat was acquired, but you do know from the grammatical tense used that it was fairly recent. With gotten, its less recent and definitely not in the present.
Or to put it another way, "got" is the verb, "ten" is the stem which indicates the tense. If you don't need to indicate that it was in the past then you don't use the "ten" suffix indicating the past tense or past imperfect/imperative.