We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in Zhang (2017) and Jung & Zhang (2018). In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod's $M_1$-topology to an $α$-stable Lévy motion, where $α$ depends on the `curvature' of the flattest points and the skewness parameter $ξ$ depends on the values of the observable at those same points. Previously, Jung & Zhang (2018) proved convergence of the one-point marginals to totally skewed $α$-stable distributions for a single symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but also allow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits we obtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show that convergence in the stronger $J_1$-topology is not possible.