As it turns out, it doesn't seem to work. The general solution for $\beta$ looks like $\beta$ = $\frac{1}{r_1^2-1}(\frac{r_1^2}{2} - \frac{1}{2} - \frac{\partial H + \partial F + \partial G}{\partial \theta_1})$, where $H$ is a function of $r_1$ and $\theta_1$, $F$ of $\theta_2$ and $\theta_1$, and $G$ of $r_2$ and $\theta_1$. Near $r_1 = 0$, for $\beta$ to be $o(r_1)$, $ \frac{\partial H + \partial F + \partial G}{\partial \theta_1}$ must be independent of $\theta_1$, but since the functions must be periodic restricted to $\theta_1$, no solution exists anywhere for all values of $\theta_1$.

The equations come from here: let $V = \beta (r_1-\frac{1}{r_1}) \partial_{r_1} + \gamma r_2 \partial_{r_2} + \alpha \partial_{\theta_1} + \mu \partial_{\theta_2}$, and $\omega = r_1 d r_1 d\theta_1 + r_2 d r_2 d\theta_2$. When is $\mathcal{L}_{V}\omega = \omega$? I checked a few times, I think the signs are correct. The idea is: try and extend the (Liouville) vector field $V_1 = (r_1-\frac{1}{r_1}) \partial_{r_1} + r_2 \partial_{r_2}$ over some points in $H_1 \cap H_2$. I don't know if the solution should have any geometric significance at all, and $V$ is indeed just a guess. It may not work.