Unfortunately I can't get explicit estimates on $\epsilon(r,Q)$, but here's an argument to show such an $\epsilon$ exists. As a notational remark, I'm going to denote the Riemannian distance between $p$ and $q$ in the metric $g$ by $d(p,q)$ and the distance from $p$ to $q$ with respect to the inner product $g(0)$ (extended in the natural way to all of $U$) by $|p-q|_g$.


Suppose that $g$ is $C^2$-scaled controlled with precision $Q$. Then, whenever $|p-q|_g$ is small, $d(p,q)$ and $|p-q|_g$ are close in $C^2$ norm. If I'm remembering the implicit function theorem correctly, it can be used to show that if two functions are close in $C^2$ norm, their level sets are close in $C^2$ norm as well. The level sets of $|p-q|_g$ are ellipsoids that are uniformly convex (by the $Q$-precision) and since any surface that is $C^2$-close to a strictly-convex surface is itself convex, for small $r$ the level set $d(p,q) = r$ is also convex.

In fact, we could have done even better. Since any surface that is $C^1$ close to a strictly-convex surface is also convex, if $g$ is merely $C^1$-scale controlled, we get the same result. It's a good question whether this can be made explicit and I'll add details if I think of anything towards that end.