“Got” vs “gotten” are both past participles of the verb “to get” and are both acceptable. Comparing them is a fairly common question (for example, see here), but the usual reply—-that it is simply an American English versus British English issue§—-is an over-simplification.
In fact “gotten” used to be the more common in both the UK and North America until well into the 19th Century. It then declined in both places, only to see a resurgence in the latter half of the 20th century. That resurgence has been stronger in the US, hence the appearance (well, the fact) that it is the AmE preferred form. But that is, as I say, only the current fashion.
So in practice, “gotten” is permissible in both AmE and BrE, although for sure it is much more likely to come across as a bit anachronistic in the latter. For myself—- I am originally from the UK but now live in the US—-other than when I was very young, I have always found “have got” a bit uncouth and I switched to “have gotten” as a young adult, and that was well before moving to the the States. That said, many of my friends (from the UK, US, and beyond) tell me my writing can come across as a bit “posh” which I guess merely confirms the common view! 😀
This blog post gives a nice explanation of the whole thing.
§With AmE preferring “gotten”, while “got” is the modern BrE choice.