Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-wise" weak equivalences of simplicial sets (in the usual sense) and cofibrations are the injections.
Is this model structure combinatorial?
That's not the Moerdijk model structure. In the Moerdijk model structure, weak equivalences and fibrations are created by the diagonal simplicial set construction. Your model structure resembles the Bousfield-Kan model structure, where weak equivalences and fibrations are defined pointwise as in simplicial sets. You rather define weak equivalences and cofibrations pointwise as in simplicial sets. This is known as the injective model structure in the diagram category of simiplicial objects simplicial sets (which are the same as bisimplicial sets). This model structure is known to be combinatorial since that of simplial sets is. See Lurie's book.
