Any FizzBuzz example almost feels too silly to deserve a serious comment. But here goes.

As you noticed, the FizzBuzz pattern has a period of 15. So if you want to go fast, just go directly to a 15x vectorized unroll with unaligned stores.

Assuming 32-byte vectors like with AVX2, you initialize two vector registers v0, v1 to hold the pattern of 15 4-byte values (the 16th value doesn't matter) corresponding to 0 through 15. You also have corresponding delta vector registers d0, d1 whose entries are +15 for the "pass-through" values and 0 for the -1/-2/-3 sentinel values. The loop kernel is just this:

    store_unaligned(p, v0)
    store_unaligned(p + 8, v1)
    v0 += d0
    v1 += d1
    p += 15

And of course you still need the remainder loop.

You could remove the unaligned stores with a monstrous lcm(15, 16) = 240x unroll (16 shifted instances of the above kernel) but you wouldn't be able to keep all the necessary loop state in registers.