I would like to check if my following sentence structure is correct.
You've got to see my enemy's face when I was winning.
You got to see my enemy's face when I was winning.
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The OP's sentence: You got to see my enemy's face when I was winning.
Grammatical: Yes but the have got form here specifically would be incorrect for this specific meaning.
Specific idiomatic usage of to get to do something: * I got to go to the sports' event last night [because I had tickets].
* He didn't get to see the movie on TV because he arrived late.
* He got to see his favorite band at the concert last night.
Meaning: to be able to do something when there is something else that could have stopped you from doing it or interfered with your doing it.
Compare the above to: to have to/have got to:
You just have to see that movie. It's great. You've just got to see that movie. It's great.
Caveat; The story of get in English is very complicated. I am only addressing the specific idiomatic usage of to get to do something as possibly confused with to have/have got to do something (meaning: must do something).
The present perfect of get to do something in AmE is: have gotten to do something.
I have not gotten to go out at night recently. [meaning: not been able to]
The present perfect of get to do something in BrE would be: to have got to do something because the Brits do not use gotten anymore.
That said, this idiom is more likely to be expressed in BrE as: to manage to do something. Manage to do something is also used in AmE.