Reentrant behavior and possible $2/3$ magnetization plateau on the double-trillium langbeinite K$_2$Ni$_2$(SO$_4$)$_3$

K$_2$Ni$_2$(SO$_4$)$_3$ is a member of the langbeinite family, consisting of two intertwined $S=1$ trillium lattices, out of which one is strongly coupled (strong-TL) and the other is weakly coupled (weak-TL). Further inter-trillium interactions give rise to a highly-frustrated Heisenberg Hamiltonian. Despite ordering at low temperatures, K$_2$Ni$_2$(SO$_4$)$_3$ lies close in parameter space to a spin-liquid region that surrounds the tetratrillium limit, where each triangle belonging to strong-TL turns into a tetrahedron by connecting to a single spin from weak-TL. Here, we compare the experimentally determined magnetization process using pulsed magnetic fields up to $40$ T with classical Monte Carlo calculations, uncovering a series of phase transitions at both low and intermediate fields. Furthermore, we reveal a signature of a $2/3$ magnetization plateau consisting of a $1/3$ phase on strong-TL and a fully polarized phase on weak-TL. Although in the classical limit no plateau is expected, we find a very prominent dome structure reflecting the tendency of the system to stabilize this particular spin configuration. The presence of a dome leads to a reentrant phenomenon in which the system recovers the Hamiltonian symmetries when increasing the magnetic field. Finally, we show that this plateau-like phase is also present in the classical Heisenberg model on the single trillium and tetratrillium lattices, indicating its possible presence in the large family of double-trillium langbeinite compounds. Our findings motivate future studies on the presence of the plateau phase in the quantum limit of both trillium and double-trillium materials within the langbeinite family.