Not an answer, but an intuition for an answer (suggesting that balls should have exponentially small Cheeger constant).

Suppose that $n \geq 2$. The volume of $B^n(R)$, the ball of radius $R$ in $n$-dimensional hyperbolic space, is basically some constant (depending on $n$) times $e^{(n - 1)R}$. In dimension one ($n = 1$) we instead have some constant times $R$.

Suppose now that $M$ is a closed connected hyperbolic $n$-manifold. Pick a basepoint $x$ in $M$ and pick a graph $\Gamma$ in $M$ that nicely carries the fundamental group of $M$. Lift all of this to the universal cover $M'$ to get a graph $\Gamma'$ which is quasi-isometric to $n$-dimensional hyperbolic space. So $\Gamma'$ will be expansive, and will be roughly as expansive as balls in the ambient hyperbolic space.

So we can move interchangably between large balls in the graph $\Gamma'$ and large balls in hyperbolic space. Let's assume that $n$ is at least three. Now consider $B^n(R)$, the ball of radius $R$ in $n$-dimensional hyperbolic space. This is cut exactly in half by its "equatorial disk", a copy of $B^{n-1}(R)$. The volume of half of $B^n(R)$ is roughly $e^{(n-1)R}$. The volume of the equatorial disk is roughly $e^{(n-2)R}$. So the ratio is $e^{-R}$, as suggested by the original question.

To make these examples work, we needed a notion of dimension. So it is not exactly clear how this generalises.

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