You have made a thorough hash of the Campbell identity in your "SU(2)" calculation, by ignoring the summation convention. Of course, you need not have used it, since Pauli vectors are explicitly calculable, so that 
$$
e^{-i\theta\hat{n}\cdot\sigma/2}\sigma_{j}e^{i\theta\hat{n}\cdot\sigma/2}\\  = (\cos(\theta/2)-i \sin(\theta/2)~\hat{n}\cdot\sigma)~\sigma_{j} ~(\cos(\theta/2)+i \sin(\theta/2)~\hat{n}\cdot\sigma)\\
= \cos^2(\theta/2)~\sigma_j -in^i[\sigma_i,\sigma_j]\cos(\theta/2)\sin(\theta/2)+\sin^2(\theta/2) n_in_k\sigma_i\sigma_j\sigma_k \\
=  \cos(\theta)\sigma_j+ n_i\epsilon^{ijk}\sigma_k \sin\theta+2\sin^2(\theta/2)n_j~\hat{n}\cdot\sigma.
$$
Can you take it from here?