I'm useless on formal grammar, but:
I went to the shop, I got milk.
I would have gone to the shop, I would have gotten milk.
We (Ireland) would never say "would have got milk", that would be incorrect here. We'd also never say "I gotten milk" - got and gotten are different tenses of the same verb.
When writing formally, there's often a better choice than any tense of "get", but that applies equally to got and gotten.