Why does strong Lewis acid-strong Lewis base interactions prevail over hard-soft acid-base interactions?
The Klopman-Salem equation accounts for the energy gain (or loss) when the orbitals of one reactant overlap with those of another. The equation is

$$\Delta E=-\sum_{ab}(p_a+p_b)\beta_{ab}S_{ab}+\sum_{k&lt;l}\frac{Q_kQ_l}{\varepsilon R_{kl}}+\sum_{r}^{\text{occ}}\sum_{s}^{\text{unocc}}-\sum_{s}^{\text{occ}}\sum_{r}^{\text{unocc}}\frac{2(\sum_{ab}c_{ra}c_{sb}\beta_{ab})^2}{E_r-E_s}$$
where $p_a$ and $p_b$ are the electron population in the atomic orbitals $a$ and $b$, $\beta$ and $S$ are the resonance and overlap integrals, $Q_k$ and $Q_l$ are the local charges on atoms $k$ and $l$, $c_{ra}$ is the coefficient of atomic orbital $a$ in molecular orbital $r$, where $r$ refers to the molecular orbitals on one molecule and $s$ refers to those on the other, $E_r$ is the energy of molecular orbital $r$ and $E_s$ is the energy of molecular orbital $s$.

The first term is a closed-shell repulsion term, the second term is the Coulombic repulsion or attraction, and the third term is the interaction of all the filled orbitals with all the unfilled of correct symmetry. This equation was derived using perturbation theory being the first term is a first-order perturbation term and the third the second-order perturbation term.

Now, the HSAB theory is affected by charge and orbital interactions. The h-h interactions is mainly electrostatic in nature, while in a s-s interaction the orbitals involved are close in energy. With the Klopman-Salem equation we can see that the second term is the hard contribution and the third is the soft contribution. There is a simplified form, which accounts only for frontier orbitals, that is
$$\Delta E=\frac{Q_{\text{Nu}}Q_{\text{El}}}{\varepsilon R}+\frac{2(c_{\text{Nu}}c_{\text{El}}\beta)^2}{E_{\text{HOMO}(\text{Nu})}\pm E_{\text{LUMO}(\text{El})}}$$
So, back to the question, a hard molecule or ion is characterized by a large HOMO-LUMO gap (definition of hardness) and the second term, in the simplified version, is very small, so the contribution is mainly electrostatic, i.e., the more charged the sites in the molecule/ion the greater will be the stabilization (or repulsion, in case of like charges). The electric field of the two parts in the molecules/ions would diminish $R$ and the term would be very large. In the case of soft-soft interactions, the orbital term is the dominant because the HOMO-LUMO gap is smaller.

But sure, the interaction has to be between to molecules or ions, not a single specie. You can apply this to nucleophiles and electrophiles as well. A hard nucleophile (negative charge localized and low-energy HOMO) would tend to react with a hard electrophile (positive charge localized and high-energy LUMO), leading to a great stabilization by the electrostatic term. Meanwhile a soft nucleophile (high-energy HOMO) would react with a soft electrophile (low-energy LUMO), stabilized by the orbital term, because $E_{\text{HOMO}(\text{Nu})}-E_{\text{LUMO}(\text{El})}$ is small.

If it is necessary, I can edit with a MO diagram to see this more visually.