I don't know of any paint that would hold up in a sink. Not knowing the material your sink is made from (plastic, porcelain, metal), the only thing "I" can think of is to have the sink reglazed. That would be the same process that is used for reglazing a bathtub.
Here's a 10 minute video how one method is done. If it works in the bathtub, it should work in a sink just as well.
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2019 Montana High Country 375FL
2014 Chevy Silverado Duramax, 6.6L Dually