When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\mathcal{N}$. I have a proof written down somewhere, but if I recall correctly, the basic idea is to choose generating (acyclic) cofibrations $I$ ($J$) for $\mathcal{M}$ and use these to write down $\mathcal{C}$ as the preimage under a suitable left adjoint $$\mathcal{M} \to \prod_{f \in I} \mathcal{N}^{\cdot \to \cdot} \times \prod_{g \in J} \mathcal{N}^{\cdot \to \cdot}$$ where the two $\mathcal{N}^{\cdot \to \cdot}$s are equipped with (different) suitable model structures cooked up to produce the class of morphisms under consideration. Then you can apply the result of M. Makkai, J. Rosický, Cellular categories, to conclude that $\mathcal{C}$ is the left class of a weak factorization system.

The weak factorization systems of an injective model category structure on diagrams can be seen as a special case of this construction, so you are going to need to do something relatively difficult to prove this.

When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\mathcal{N}$. I have a proof written down somewhere, but if I recall correctly, the basic idea is to choose generating (acyclic) cofibrations $I$ ($J$) for $\mathcal{M}$ and use these to write down $\mathcal{C}$ as the preimage under a suitable left adjoint $$\mathrm{Fun^L}(\mathcal{M}, \mathcal{N}) \to \prod_{f \in I} \mathcal{N}^{\cdot \to \cdot} \times \prod_{g \in J} \mathcal{N}^{\cdot \to \cdot}$$ where the two $\mathcal{N}^{\cdot \to \cdot}$s are equipped with (different) suitable model structures cooked up to produce the class of morphisms under consideration. Then you can apply the result of M. Makkai, J. Rosický, Cellular categories, to conclude that $\mathcal{C}$ is the left class of a weak factorization system.

The weak factorization systems of an injective model category structure on diagrams can be seen as a special case of this construction, so you are going to need to do something relatively difficult to prove this.