@Hannes I was thinking of an a priori argument (assuming you have bounds for $u$ do they also hold for $v$) but I have now updated the question to ensure the regularity for $u$

@sorrymaker symmetry is not the issue for me, I am fine with assuming it (I have updated the question accordingly)

@Corbennick: Let $\tau_r,\delta_r>0$ be such that $|a-\partial_2 F_r(x,y)|\leq a/2$ for all $(x,y)\in\overline{B(0,\tau_r)}\times\overline{B(0,\delta_r)}$ and let $\varepsilon_r<\tau_r$ such that $|F_r(x,0)|<\delta_r/2$ for all $x\in B(0,\varepsilon_r)$ (all possible due to the assumptions). Then, basically, $\varepsilon_r$ is the size of the open set $U_r$. If $\varepsilon_r$ and, in consequence $\tau_r$, can be chosen independently of the parameter $r$, we can get a uniform neighborhood $U_r$