I suspect that you already have one, but here is a proof. I will assume that MM and N are combinatorial and that M is left proper (otherwise, I don't think that the literature contains a general construction of the left Bousfield localizations of M by any small set of maps). Everything needed for a quick proof is available in Appendix A of
J. Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, 2009.
First, there exists a cofibrant resolution functor Q in M which is accessible: the one obtained by the small object argument (as accessible functors are closed under colimits, it is sufficient to know that HomM(X,−) is an accessible functor for any object X in M, which is true, as N is combinatorial). Let W be the class of maps f of M such that L(Q(f)) is a weak equivalence in N. As N is combinatorial, the class of weak equivalences of N is accessible see Corollary A.2.6.9 in loc. cit. Therefore, by virtue of Corollary A.2.6.5 in loc. cit, the class W is accessible. To Prove what you want, it is sufficient to check that M, W and C={cofibrations of M} satisfy the conditions of Proposition A.2.6.8 in loc. cit. The only non trivial part is the fact that the class CcapW satisfies all the usual stability properties for a class of trivial cofibrations (namely: stability by pushout, transfinite composition). That is where we use the left properness. For instance, if AtoB is in W and if AtoA′ is a cofibration of M, we would like the map A′toB′=A′amalgAB to be in W as well. This is clear, by definition, if A, A′ and B are cofibrant. For the general case, as M is left proper, B′ is (weakly equivalent to) the homotopy pushout A′amalghAB, and as left derived functors of left Quillen functors preserve homotopy pushouts, we may assume after all that A, A′ and B are cofibrant (by considering the adequate cofibrant resolution to construct the homotopy pushout in a canonical way), and we are done. The case of transfinite composition is similar.
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