Tada! That's very close to what I had in mind. The only difference: you can fizz on 4924, buzz on 4210, none on 34cb, then always \n. Also, you have 3 ops to calculate the rotation, and that can be replaced by (0x8001 * x) >> 1 (for uint16_t x), but with tweaking of the masks because it's rotating right.
Congrats!