  We perform an extensive numerical study of the disordered Poland-Scheraga
(PS) model for DNA denaturation in which self-avoidance is completely taken
into account. In complement to our previous work, we focus here on the finite
size scaling in terms of pseudo-critical temperatures. We find notably that the
mean value and the fluctuations of the pseudo-$T_c$ scale with the same
exponent, the correlation length exponent $\nu_r$ (for which we provide the
refined evaluation $\nu_r=2.9 \pm 0.4$). This result (coherent with the typical
picture that describes random ferromagnets, when disorder is relevant) is at
variance with numerical results reported in the literature for the PS model
with self-avoidance, leading to an alternative scenario with a pseudo first
order transition. We moreover introduce a crossover chain length $N^*$, which
we evaluate, appropriate for characterizing the approach to the asymptotic
regime in this model. Essentially, below $N^*$, the behaviour of the model in
our study could also agree with such alternative scenario. Based on an
approximate prediction of the dependence of $N^*$ on the parameters of the
model, we show that following the choice of such parameters it could be not
possible to reach the asymptotic regime in practice. In such context it becomes
then possible to reconcile the apparently contradictory numerical studies.
