I will try to give you an answer, though with a slightly different approach. There are several things to be agreed upon before entering into this discussion.
First, what is "chaotic dynamics"? Although, to my knowledge, there is no a universally accepted mathematical definition of chaos, we do have ways to "classify" what a chaotic system would be. We could, among other things, require the system to be topologically transitive: let $f:X\rightarrow X$ be a continuos map, we say that $f$ is topologically transitive if, for every pair of non-empty open sets $A,B\subset X$, there exists an integer $n$ such that
$$f^n(A)\cap B\neq\emptyset.$$
Second, what is "integrability"? This is one mathematical concept that has been approached in several different ways.
Dynamicists and mathematicians who work on billiards would most likely prefer the definition that goes along the lines: a billiard modelled on a table with boundary $\partial \Omega$ is said to be integrable if $\Omega\subset \mathbb{R}^2$ can be foliated by closed curves s.t. every billiard orbit that is tangent to one such curve, at some of its bouncing trajectories, will always be tangent to this curve for all its future and past bouncing trajectories. We call on such a curve a caustic. In the case of the billiard on the circular table for example, it is an interesting planar geometry exercise to show that all concentric circles are caustics, and hence, the billiard dynamics is integrable.
On the other hand, mathematicians that work on differential geometry would refer to an integrable system in a seemingly different way. Let $(M^{2n},\omega)$ be a symplectic manifold, that is, $\omega$ is a non-degenerate closed 2-form. Consider any Hamiltonian $H:M\rightarrow\mathbb{R}$ ("Hamiltonian" here is just jargon meaning a smooth function), we can then define the Hamiltonian vector field $X_H$ as the unique vector field s.t. $i_{X_H}\omega=dH$ and the flow of this vector field gives you a time-dependent dynamical system $\phi^t: M\rightarrow M$. We say that system is integrable if there exists $n-1$ functions $f_1,\ldots,f_{n-1}:M\rightarrow \mathbb{R}$ s.t $\{H,f_i\}=0$ and $\{f_i,f_j\}=0$ for every $i,j$, where $\{\cdot,\cdot\}$ denotes the Poisson bracket. It turns out that trajectories on integrable systems can be packed in families of half-dimensional tori (at least in the regular strata), some of them foliated by closed trajectories, whereas the others contain dense trajectories. But the moral is, that for integrable systems, trajectories are restrained to live inside these half-dimensional tori. Now, if you take two different such tori and consider non-intersecting tubular neighborhoods of each as your subsets $A$ and $B$, you can show that $\phi^t(A)\cap B=\emptyset$ for every $t$, so the system cannot be chaotic.
Going back to your billiard problem, we could try to talk about integrabilty in the "geometers" sense. Of course, we have the caveat that the billiard system is not smooth, so it will not come as the flow of some Hamiltonian, not a smooth Hamiltonian at least. But here is when things get interesting, for $\varepsilon>0$ consider the Hamiltonian $H_{\varepsilon}:T^*\mathbb{R}^2\cong \mathbb{R}_x^2\times \mathbb{R}_y^2\rightarrow \mathbb{R}$ defined as
$$H_{\varepsilon}(x_1,x_2,y_1,y_2)=\frac{1}{2}||y||^2+\frac{\varepsilon}{1-||x||^2}.$$
For any point $x$ s.t. $||x||<1$, we have that the curve defined by $\phi^t(x)$ for every $t\in \mathbb{R}$ is contained within the unit disk, and as $\varepsilon \rightarrow 0$, this trajectory resembles more and more an actual billiard trajectory on the disk, meaning it converges to a billiard trajectory when $\varepsilon \rightarrow 0$. So we ca use this perturbed system to describe the billiard dynamic on the circle. Now if you take the angular momentum Hamiltonian
$$J(x,y)=x\times y,$$
you can show that $\{H_{\varepsilon},J\}=0$ for every $\varepsilon>0$. So we have an integrable system, and we can see that in this sense, the billiard dynamics is also integrable.
