I am trying to understand the following fact:
Given a compact Riemannian closed 3-manifold $M$ such that there is a surjection $\phi:\pi_1(M)\rightarrow \mathbb Z *\mathbb Z$ with non-trivial kernel then the injectivity radius at every point of the cover of $M$ corresponding to $ker(\phi)$ has an upper bound, i.e. there are no arbitrarily large isometrically embedded balls in the cover.
If $N$ is a connected Riemannian $n$-manifold which is not diffeomorphic to $R^n$, then the injectivity radius of $M$ is finite (pretty much by the definition), hence, has an upper bound.
As for your question about existence of an upper bound on $InRad(x)$, $x\in N$ (where $N$ is the covering space), it is again a general fact:
Suppose that $N$ is a Riemannian $n$-manifold which admits a cocompact isometric group action and $N$ is not diffeomorphic to $R^n$. Then $$ \sup_{x\in N} InRad(x) <\infty. $$ This follows from the fact that for every compact $C\subset N$, $$ \sup_{x\in C} InRad(x) <\infty. $$