I think there is a misunderstanding here on what is the expansion parameter. 

Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.

Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.

The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.