In this letter we show how the problem of assessing structural local stability of BDC-decomposable systems, left open in recent literature, can be solved via convex optimisation. First we give a simple test, based on a sufficient condition, that requires checking the strict co-positivity of a multivariate polynomial. Then we provide a stronger test, based on a necessary and sufficient condition, which can be numerically implemented via LMI-based convex optimisation. The proposed approach certifies the structural stability of non-trivial systems, including a biological network discussed in the literature.