Additionally, have can be a main verb or an auxiliary to a different main verb. The have in I have sneezed is different from the one in I have a bucket.
The have in I have gotten a bucket is used as an auxiliary, but in I have a bucket it’s the main verb usage. So gotten is not just something you can optionally add to have, it’s actually a whole different construction either way.
Anyway, I think the difference between have and have gotten is that have describes a state of currently possessing something, while have gotten focuses on a past event of acquiring it and only implies you still have it right now.