Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$ such that ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$? Or even $\frac{f}{r^2}$ where $f \in C^{\infty}(S^2)$ satisfying $\int_{S^2} f \neq 0$? Is there an easy way to find an example?

$g$ is asymptotically flat if there exists coordinates such that $g_{ij} - \delta_{ij} = O_{2}(r^{-1})$. 

It is easy to see that $g$ cannot be conformal to $g_{euc}$ since ${\rm Ric}_{r\phi}$ will be equal to $\partial_{\phi} A$ for some function $A$ on the sphere, and so integrates to $0$.