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-BCG*. The weak lensing profiles Delta Sigma of the two cluster subsamples exhibit different slopes on scales below 1 h(-1) Mpc. To interpret such discrepancy, we perform a comprehensive Bayesian modelling of the two Delta Sigma 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 x 10(14) h(-1) M-circle dot, but the concentration of the low-M-BCG* clusters is 5.87(-0.60)(+0.77), similar to 1.5 sigma smaller than that of their high-M-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-BCG*, low-concentration clusters are similar to 10 per cent higher than that of the high-M-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) kpc, the dark matter mass within similar to 200 h(-1) kpc, and the cosmic overdensity on scales above 10 h(-1) Mpc, enabling a key observational test of theories of co-evolution between massive clusters and their central galaxies.