Scales for co-compact embeddings of virtually free groups

Let $\Gamma$ be a group which is virtually free of rank at least 2 and let $\mathcal{F}_{td}(\Gamma)$ be the family of totally disconnected, locally compact groups containing $\Gamma$ as a co-compact lattice. We prove that the values of the scale function with respect to groups in $\mathcal{F}_{td}(\Gamma)$ evaluated on the subset $\Gamma$ have only finitely many prime divisors. This can be thought of as a uniform property of the family $\mathcal{F}_{td}(\Gamma)$.