We explore higher-order topological superconductivity in an artificial Dirac material with intrinsic spin-orbit coupling, which is a doped Z(2) topological insulator in the normal state. A mechanism for superconductivity due to repulsive interactions, pseudospin pairing, has recently been shown to naturally result in higher-order topology in Dirac systems past a minimum chemical potential [T. Li et al., 2D Mater. 9, 015031 (2022)]. Here we apply this theory through microscopic modeling of a superlattice potential imposed on an inversion-symmetric hole-doped semiconductor heterostructure, known as hole-based semiconductor artificial graphene, and extend previous work to include the effects of spin-orbit coupling. We find that spin-orbit coupling enhances interaction effects, providing an experimental handle to increase the efficiency of the superconducting mechanism. We show that the phase diagram of these systems, as a function of chemical potential and interaction strength, contains three superconducting states: a first-order topological p + ip state, a second-order topological spatially modulated p + i tau p state, and a second-order topological extended s-wave state s(tau). We calculate the symmetry-based indicators for the p + i tau p and s(tau) states, which prove these states possess second-order topology. Exact diagonalization results are presented which illustrate the interplay between the boundary physics and spin-orbit interaction. We argue that this class of systems offers an experimental platform to engineer and explore first- and higher-order topological superconducting states.