The variation is not nice, and may not be very helpful. The best I can make of it is this:
For a curve $C$ such that $r \circ C$ is minimized by a unique $t$, the first variation is $-(\nabla r(C(t)) \cdot \gamma(t)) r(C(t))^2$. For curves on which $r$ does not have a unique minimum, the variation does not exist, except for very special $\gamma$, because the upper and lower limits differ.
Assuming the objective is to find curves, with given endpoints, that maximize $\min(r\circ C)$, the only conclusion that can be drawn is that if $r$ has a unique minimum on sucb a curve, other than at an endpoint, then it must be at a stationary point of $r$.
That this conclusion is not very specific should not come as a big surprise, as generally $\min(r\circ C)$ is maximized by a huge family of curves. If we call the endpoints $a$ and $b$, then basically the problem is to find the largest $m$ such that $a$ and $b$ are in the same path-component of $r^{-1}([m, \infty)).$ For such $m$, any curve in $r^{-1}([m, \infty))$ between $a$ and $b$ minimizes $F$.
To make this easier to visualize, restrict to two dimensions and take $r(x, y) = xy$. Let $a = (1,1)$ and $b = (-1,-1)$. For $m > 0$, the set where $xy \ge m$ is disconnected and $a$ and $b$ lie in different components. The set where $xy \ge 0$ consists of the first and third quadrant, including axes. Any curve within this set that connects $a$ and $b$ minimizes $F$ and the only thing they have in common is that they pass through $(0, 0)$, the only stationary point of $r$.