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 
$$
e^{-i\theta\hat{n}\cdot\sigma/2}\sigma_{j}e^{i\theta\hat{n}\cdot\sigma/2}\\  = (\cos(\theta/2)-i \sin(\theta/2)~\hat{n}\cdot\sigma)~\sigma_{j} ~(\cos(\theta/2)+i \sin(\theta/2)~\hat{n}\cdot\sigma)\\
= \cos^2(\theta/2)~\sigma_j -in^i[\sigma_i,\sigma_j]\cos(\theta/2)\sin(\theta/2)+\sin^2(\theta/2) n_in_k\sigma_i\sigma_j\sigma_k \\
=  \cos\!\theta~\sigma_j+ n_i\epsilon^{ijk}\sigma_k \sin\theta+(1-\cos\!\theta) ~n_j~\hat{n}\cdot\sigma,
$$
cf. [Rodrigues' rotation formula](https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula), i.e. adjoint SU(2) action on generators rotates them like vectors under SO(3).

Can you take it from here?