The Weak $(2,2)$-Conjecture is a graph labelling problem asking whether all connected graphs of at least three vertices can have their edges assigned red labels~$1$ and $2$ and blue labels~$1$ and~$2$ so that any two adjacent vertices are distinguished either by their sums of incident red labels, or by their sums of incident blue labels. This problem emerged in a recent work aiming at proposing a general framework encapsulating several distinguishing labelling problems and notions, such as the well-known 1-2-3 Conjecture, a few of its variants, and so-called locally irregular decompositions. One further point of interest behind the Weak $(2,2)$-Conjecture is that it is weaker than the 1-2-3 Conjecture, in the sense that the latter conjecture, if proved true, would imply the former one is true too. In this work, we prove that the Weak $(2,2)$-Conjecture holds for two classes of graphs defined in terms of forbidden induced structures, namely claw-free graphs and graphs with no pair of independent edges. One main point of interest for focusing on such classes of graphs is that the 1-2-3 Conjecture is not known to hold for them. Also, these two classes of graphs have unbounded chromatic number, while the 1-2-3 Conjecture is mostly understood for classes with bounded and low chromatic number.