  Spin-singlet orders are studied for the antiferromagnetic Heisenberg model
with spin $S$>1/2 on a breathing pyrochlore lattice, where tetrahedron units
are weakly coupled and exchange constants have two values $0<J' \ll J$. The
ground state has a thermodynamic degeneracy at $J'$=0, and I have studied
lattice symmetry breaking associated to lifting this degeneracy. Third-order
perturbation in $J'$ for general spin $S$ shows that the effective Hamiltonian
has a form of three-tetrahedron interactions of pseudospins $\tau$, which is
identical to that previously derived for $S$=1/2, and I have calculated their
matrix elements for general $S$. For this effective Hamiltonian, I have
obtained its mean-field ground state and investigated the possibility of
lattice symmetry breaking for the cases of $S$=3/2 and $1$. In contrast to the
$S$=1/2 case,$\tau$'s response to conjugate field has a $Z_3$ anisotropy in its
internal space, and this stabilizes the mean-field ground state. The mean-field
ground state has a characteristic spatial pattern of spin correlations related
to the lattice symmetry breaking. Spin structure factor $S(q)$ is calculated
and found to have symmetry broken parts with amplitudes of the same order as
the isotropic part.
