Consider the infinite 3-regular tree. Pick a vertex $C$, the "center". For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced subgraph on the set of vertices given by this ball. The finite tree $T_L$ rooted at $C$ has $L+1$ layers or generations. It has $3\times 2^L-2$ vertices and they all have degree $3$ except the $3\times 2^{L-1}$ leaves in the top layer. My question is:

Q1:Is it possible to add $3\times 2^{L-1}$ edges between the leaves in order to get a graph $G_L$ with spanning tree $T_L$ such that $G_L$ is vertex-transitive? (Edit: now solved in the negative, see below)

For $L=1$ and $L=2$, I can do that which gives me the **tetrahedron** and the **Petersen graph** respectively.
Does this kind of problem arise in geometric group theory and the study of locally finite graphs?

Also, when looking at OEIS A032355, the relevant numbers of vertex-transitive graphs seems abnormally low. Even more surprising, they immediately precede a big surge (look at the values for $n=2,5,11,23,47,95$ in that list). Is there a known explanation for this phenomenon?

**Edit in view of the awesome answers by Aaron and Gordon:**
I still have to digest the theory around Moore graphs and bound. My motivation for this question comes from statistical mechanics on the widest possible class of lattices. In particular it has to do with the question of defining **"periodic boundary conditions"** for given finite subsets.
In this regard, an equally useful (for me) weakening of my question is as follows:

Q2:Is there a constant $c\in (0,1)$ independent of $L$ such that one can build $G_L$ as above to make the size of the orbit of the center $C$ no smaller than $c\times(3\times 2^{L}-2)$?

In other words I want to make this orbit of macroscopic size. The orbit is of course wih respect to the action of the automorphism group of $G_L$. A quick remark is that if one adds no edges, i.e., one takes $T_L$ itself, this orbit is the singleton $\{C\}$ and the automorphims group is a wreath product $\mathbb{Z}_2\wr\cdots\wr\mathbb{Z}_2\wr\mathbb{Z}_3$ with $\mathbb{Z}_2$ appearing $L-1$ times.