I know that normally in American English we use "gotten" after has/have but in this type of construction, do Americans say "got" or "gotten"? As in
"I have got to sing"
"I have gotten to sing"?
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Exactly as Colin Fine said, both could be used and they have very different meanings in American English.
I have got to sing means I must sing—I have an obligation or a need or a drive to sing. It is a present-tense expression of need.
I have gotten to sing means I have had the opportunity to sing. Note that this is the past tense now—we are describing an opportunity that existed at one point in the past.
I got one answer from MarcinManhattan on my other thread which I am adding here. He said the same as you both said. "I have got to go to the party" means "I have to go to the party" (present time). "I have gotten to go to the party" means "I have been able to go to the party" (present perfect).