We introduce a scheme of a photonic coupler built of two parallel topological-insulator slab waveguides with the intrinsic Kerr nonlinearity, separated by a lattice spacing. Josephson oscillations (JOs) of a single edge quasi-soliton (QS) created in one slab, and of a pair of QSs created in two slabs, are considered. The single QS jumping between the slabs is subject to quick radiative decay. However, the JOs of the co-propagating QS pair may be essentially more robust, as one QS absorbs dispersive waves emitted by the other. The most robust JO regime is featured by the pair of QSs with phase shift π between them.