Twisted endoscopic character relation for toral supercuspidal L-packets of classical groups

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's construction of the local Langlands correspondence for toral supercuspidal representations of quasi-split symplectic or special orthogonal groups coincides with Arthur's. The strategy is to emulate Kaletha's proof of the standard endoscopic character relation in the twisted setting by appealing to Waldspurger's framework ``l'endoscopie tordue n'est pas si tordue''.