Density matrix embedding theory (DMET) is a relatively new technique for the calculation of strongly correlated systems. Recently, block product DMET (BPDMET) was introduced for the study of spin systems such as the antiferromagnetic J1−J2 model on the square lattice. In this paper, we extend the variational Ansatz of BPDMET using spin-state optimization, yielding improved results. We apply the same techniques to the Kitaev-Heisenberg model on the honeycomb lattice, comparing the results when using several types of clusters. Energy profiles and correlation functions are investigated. A diagonalization in the tangent space of the variational approach yields information on the excited states and the corresponding spectral functions.