JavaScript (ES6), 82 bytes

Saved 2 bytes thanks to a suggestion by @emanresuA

Expects and returns -1 for Fizz, -2 for Buzz, -3 for FizzBuzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>-!(++i%3)-2*!(i%5)||i)].sort()+''==a?b:f(a,++k) 

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JavaScript (ES6), 82 bytes

Saved 2 bytes thanks to a suggestion by @emanresuA

Expects and returns -1 for Fizz, -2 for Buzz, -3 for FizzBuzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>-!(++i%3)-2*!(i%5)||i)].sort()+''==a?b:f(a,++k) 

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The underlying algorithm is the same as in my answer to the twin challenge.

JavaScript (ES6), 84 bytes

Expects and returns -1 for FizzBuzz, -2 for Fizz, -3 for Buzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>++i%3?i%5?i:-3:-2^i%5<1)].sort()+''==a?b:f(a,++k) 

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JavaScript (ES6), 82 bytes

Saved 2 bytes thanks to a suggestion by @emanresuA

Expects and returns -1 for Fizz, -2 for Buzz, -3 for FizzBuzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>-!(++i%3)-2*!(i%5)||i)].sort()+''==a?b:f(a,++k) 

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JavaScript (ES6), 85 bytes

Expects and returns -1 for Fizz, -2 for Buzz, -3 for FizzBuzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>++i%3?i%5?i:-2:i%5?-1:-3)].sort()+''==a?b:f(a,++k) 

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JavaScript (ES6), 84 bytes

Expects and returns -1 for FizzBuzz, -2 for Fizz, -3 for Buzz.

f=(a,i=k=0)=>[...b=a.sort().map(_=>++i%3?i%5?i:-3:-2^i%5<1)].sort()+''==a?b:f(a,++k) 

Try it online!