The thing with these Garifuna is that they are no more or less "mixed" than many people in Jamaica, Trinidad, etc.
That they object to using "black" when "Black" is generally a catch all for those of African Descent, which clearly these folks are (more so if you look at the culture) shows just how divided the black house is. Of course some of them, just like some Jamaicans, and African-Americans are not black and/or are more mixed than others.
My suggestion: They should check the "black" box and
also the other box and put in their ethnic identity. Just like there are Gullahs in the US African-American population.