We continue to test our method of Pauli-Villars regularisation on QED. Ultimately we hope to apply the method to the problem of deriving non-perturbative solutions for QCD. Developing it pursues the goal of working out a renormalisation scheme which preserves Lorentz symmetries and gauge invariance in the framework of light-cone Hamiltonian approach. In a previous work a non-perturbative solution, developed as a Fock-state expansion and truncated up to one electron - one photon states, gave a good agreement with the Schwinger term of the usual Feynman series. In this paper we add one electron and two photon terms to the eigenstate. We expect to obtain good agreement with the next order contribution to the electron's anomalous magnetic moment - the Sommerfield-Peterman term. In addition, these calculations more strictly test the proposed prescription for new singularities which arise due to the indefinite metric.