In Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure a BCH formula is given which works for an arbitrary faithful representation of SU(2). Now, is there an equivalent relation for the Zassenhaus formula?
More explicitly, can you write $\exp(it \hat n \cdot \vec\sigma)$ as a product of exponentials of the Pauli matrices? As argued in the linked post, this would give a relation that works for any faithful representation of SU(2).