ABSTRACT Concentration is one of the key dark matter halo properties that could drive the scatter in the stellar-to-halo mass relation of massive clusters. We derive robust photometric stellar masses for a sample of brightest central galaxies (BCGs) in SDSS redmapper clusters at 0.17 < z < 0.3, and split the clusters into two equal-halo mass subsamples by their BCG stellar mass $M_*^{\mathrm{BCG}}$. The weak lensing profiles ΔΣ of the two cluster subsamples exhibit different slopes on scales below $1\, h^{-1}\, {\mathrm{Mpc}}$. To interpret such discrepancy, we perform a comprehensive Bayesian modelling of the two ΔΣ profiles by including different levels of miscentring effects between the two subsamples as informed by X-ray observations. We find that the two subsamples have the same average halo mass of $1.74\times 10^{14}\, h^{-1}\, \mathrm{M}_{⊙ }$, but the concentration of the low-$M_*^{\mathrm{BCG}}$ clusters is $5.87_{-0.60}^{+0.77}$, ∼1.5σ smaller than that of their high-$M_*^{\mathrm{BCG}}$ counterparts ($6.95_{-0.66}^{+0.78}$). Furthermore, both cluster weak lensing and cluster-galaxy cross-correlations indicate that the large-scale bias of the low-$M_*^{\mathrm{BCG}}$, low-concentration clusters are ${∼}10{{\ \rm per\ cent}}$ higher than that of the high-$M_*^{\mathrm{BCG}}$, high-concentration systems, hence possible evidence of the cluster assembly bias effect. Our results reveal a remarkable physical connection between the stellar mass within $20{-}30\, h^{-1}\, {\mathrm{kpc}}$, the dark matter mass within ${∼}200\, h^{-1}\, {\mathrm{kpc}}$, and the cosmic overdensity on scales above $10\, h^{-1}\, {\mathrm{Mpc}}$, enabling a key observational test of theories of co-evolution between massive clusters and their central galaxies.