Is anyone talking about "ball bundles" of metric spaces?

Okay, this is more of a collection of rambling ideas, but it is way to long for a comment, so I will have to post it as an answer. To give a short answer, I have never seen something completely like this but I have some observations to be made, that hopefully may help.

First of all considering one of your side notes, you can can use the Hausdorff-distance to give a metric to your bundle (or rather the set of balls of different radii, which is kind of the same). One interpretation of this seems to be rather related to your problem: If $\Omega$ is a set, if you define $B_r(\Omega) := \{x\in X: d(x,\Omega)<r\}$, then the Hausdorff distance between $\Omega_1$ and $\Omega_2$ is the smallest $r$ such that $\Omega_1 \subset B_r(\Omega_2)$ and $\Omega_2 \subset B_r(\Omega_2)$. You may get in some uniqueness problems, whenever two balls around different points are the same set, but you can for example add the distance between base points to the metric. You could even try to use general sets and consider their diameter and somehow connect this idea to the similar idea of a base in topology.

Secondly, one thing that kind of bugs me is that your pushforward of balls lacks one key feature of the vector-bundle version, since it is not linear. You could do a linearized version by defining the pushforward $f_*(r,x)= (s,f(y))$ where $$ s= \lim_{h\to 0} \frac{\inf \{s>0: f(B_{rh}(x)) \subset B_s(f(x))\}}{h}.$$ This would also conform more to the fact that the pushforward is a local property and should not be influenced by points at a large distance. But then if your metric space is nice enough, lets say a manifold, you will just end up with the operator norm of the pushforward (or in case of subsets of $R^n$ just $\|Df\|$ ) which is already used in numerical analysis. But if you do not linearize then your pushforward becomes essentially incalculable compared to interval-arithmetric, since you have to do much more then calculating end points of intervals. You may be able to estimate some upper bounds, but if you have to do so anyway, you probably can use those bounds explicitely instead of hiding them in formalism.

However the idea still looks fun. I think there are a lot of slightly similar things happening in the depths geometric measure theory, but usually there one is more interested in the measure of sets than in their diameter.