My sense of things FWIMBW is that 'gotten' is something that happened in the early 1800s on this side of the pond and stayed, and never happened in Britain. It's pretty firmly in the language by the time Louisa May Alcott was writing, which says to me that it was around when she was growing up before the Civil War.

Also, we say 'I have got to go to the store', which indicates need, and 'I got to the store' (past tense); 'he got into trouble' (simple past tense) and 'he has gotten into trouble again', (a version of present tense that might be something like present continuous if I knew what tenses were being called these days.) Anyway, it seems to be a tense marker as much as anything else.