The Associated Press Stylebook (2018) has the following entry on "Quotations" under News Values on p.520:
Quotes must not be taken out of context. We do not alter quotations, even to correct grammatical errors or word usage. If a quotation is flawed because of grammar or lack of clarity, it may be paraphrased in a way that is completely true to the original quote. If a quote's meaning is too murky to be paraphrased accurately, it should not be used. Ellipses should be used rarely and must not alter the speaker's meaning.
So the journalist could paraphrase or omit the quote if they are unsure. Otherwise, they can leave it unaltered. Then, a paragraph later,
Use of regional dialects with nonstandard spellings should generally be limited to a writer's effort to convey a special tone or sense of place. In this case, as in interviews with a people not speaking their native language, it is especially important that their ideas be accurately conveyed. Always, we must be careful not to mock the people we quote.
The latter paragraph allows for slight wiggle room to represent an expression with "nonstandard spelling" in standard spelling. If "gotta" is equivalent to "got to," and "gonna" is equivalent to "going to," adjusting the spelling is allowed, but further alteration for grammar ("have got to" instead of "got to") isn't. Meanwhile, if gotta is important to capture the "tone or sense of place," use it unchanged.
So the possible responses of an AP writer would be to paraphrase the response and avoid the issue entirely, leave it unaltered (gotta), or adjust only the literal spelling to an obvious standard version (got to).