It may be only my impression, but here is what I think:

"I've got" would normally be used in Britain both for the "becoming" meaning ("I've got a bit heavier lately"), and the plain "possession" meaning ("I've got a little list").

In the States, "I've gotten" would normally be used for the continuing meaning ("I've gotten a bit heavier lately"), but the possession meaning would be expressed without "got", simply as "I have a list" or "I don't have a list".

This is what I have surmised from listening to Brits and USns, have I got it approximately right? No idea about Canadians or other species...