Perhaps techniques from the theory of pseudodifferential operators might be of help here? I have a relatively brief acquittance with the subject, but share in hope that it will be of some help to you. You can find more details in e.g. Michael E. Taylor *Partial Differential Equations part 3*, chapters 13.8 and 14.4.

If I understand correctly the symbol of the differential operator in question is:
$$
p(x,\xi)=-|\xi|^{2s}+(\xi,v(x))
$$
where $(\cdot,\cdot)$ stands for the inner product on $\mathbb{R}^{n}$. If $s\in(\frac{1}{2},1)$ then $2s>1$. By the regularity assumption on $v$, you can then use the Sobolev embedding theorem to obtain $C^{r}_{*}$-estimates on $v$, and then
$$
|D_{\xi}^{\alpha}p(x,\xi)|\leq C_{1}\langle\xi\rangle^{2s-|\alpha|}
\\\|D_{\xi}^{\alpha}p(\cdot,\xi)\|_{C^{r}_{*}}\leq C_{2}\langle\xi\rangle^{2s-|\alpha|}
$$
for some constants $C_{1},C_{2}>0$. The only nontrivial case seems to be when $|\alpha|=1$, as the $(\xi,v(x))$ term falls under further differentiations by $D_{\xi}$. Then, if again I understand correctly, the existence and regularity results in Chapter 14.4 of Taylor's book seem to apply.