The reliability study of the valence-universal coupled-cluster (VU-CC) methods [Jankowski et al., J. Chem. Phys. 97, 7600 (1992)] is continued by considering approximations involving one-, two-, and three-electron components of the cluster operator S, S similar or equal to S-1 + S-2 + S-3. The explicit equations determining the cluster amplitudes for the quadratic VU-CCSDT approximation were derived and applied to the low lying quasidegenerate states of the model system consisting of two slightly stretched, interacting hydrogen molecules. The degree of quasidegeneracy of the two lowest, totally symmetric, singlet levels of this system may be varied over a wide range by changing its geometry. Along with the quadratic VU-CCSDT theory its linear version (VU-LCCSDT) is also examined. It is found that the effect of the S-3 clusters is significant and is mainly due to the spectator part of this operator. The inclusion of this operator increases the number of physically meaningful multiple solutions of the equations for the cluster amplitudes. It also considerably improves the accuracy of the resulting energies, both in the strongly and weakly quasidegenerate regions, but not in the intermediately quasidegenerate region. It is shown that this behavior is a manifestation of an avoided crossing of the lowest excited states of the parent system in the valence-universal genealogy.