Cluster weak lensing is a sensitive probe of cosmology, particularly the amplitude of matter clustering $\sigma_8$ and matter density parameter $\Omega_m$. The main nuisance parameter in a cluster weak lensing cosmological analysis is the scatter between the true halo mass and the relevant cluster observable, denoted $\sigma_{\ln M c}$. We show that combining the cluster weak lensing observable $\Delta \Sigma$ with the projected cluster-galaxy cross-correlation function $w_{p,cg}$ and galaxy auto-correlation function $w_{p,gg}$ can break the degeneracy between $\sigma_8$ and $\sigma_{\ln M c}$ to achieve tight, percent-level constraints on $\sigma_8$. Using a grid of cosmological N-body simulations, we compute derivatives of $\Delta \Sigma$, $w_{p,cg}$, and $w_{p,gg}$ with respect to $\sigma_8$, $\Omega_m$, $\sigma_{\ln M c}$ and halo occupation distribution (HOD) parameters describing the galaxy population. We also compute covariance matrices motivated by the properties of the Dark Energy Suvery (DES) cluster and weak lensing survey and the BOSS CMASS galaxy redshift survey. For our fiducial scenario combining $\Delta \Sigma$, $w_{p,cg}$, and $w_{p,gg}$ measured over $0.3-30.0 h^{-1} \mathrm{Mpc}$, for clusters at $z=0.35-0.55$ above a mass threshold $M_c\approx 2\times 10^{14} h^{-1} M_\odot$, we forecast a $1.4$% constraint on $\sigma_8$ while marginalizing over $\sigma_{\ln M c}$ and all HOD parameters. Reducing the mass threshold to $1\times 10^{14} h^{-1} M_\odot$ and adding a $z=0.15-0.35$ redshift bin sharpens this constraint to $0.8$%. The small scale $(r_p < 3.0 h^{-1} \mathrm{Mpc})$ ‘‘mass function’’ and large scale $(r_p > 3.0 h^{-1} \mathrm{Mpc})$ ‘‘halo-mass cross-correlation’’ regimes of $\Delta \Sigma$ have comparable constraining power, allowing internal consistency tests from such an analysis.