We looked into this with Greta and Damir in this paperOn the largest Kronecker and Littlewood–Richardson coefficients. Unfortunately, other than the Harris-WillenbringHarris–Willenbring identity we didn't find much. For our bounds we even had to invent a new (not too difficult) LR-identity which follows from the skew Cauchy identity (it's Lemma 5.1 in the paper): $$ \sum_{\lambda\vdash n} \left( c^{\lambda}_{\mu\nu}\right)^2 \, = \, \sum_{\alpha, \beta, \gamma, \delta} c^{\mu}_{\alpha\gamma} c^{\mu}_{\alpha \delta} c^{\nu}_{\beta\gamma} c^{\nu}_{\beta \delta} $$$$ \sum_{\lambda\vdash n} \left( c^{\lambda}_{\mu\nu}\right)^2 \, = \, \sum_{\alpha, \beta, \gamma, \delta} c^{\mu}_{\alpha\gamma} c^{\mu}_{\alpha \delta} c^{\nu}_{\beta\gamma} c^{\nu}_{\beta \delta}. $$
