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We present KaRMMa 2.0, an updated version of the mass map reconstruction code introduced in Fiedorowicz et al. (2022). KaRMMa is a full-sky Bayesian algorithm for reconstructing weak lensing mass maps from shear data. It forward-models the convergence field as a realization of a lognormal field. The corresponding shear map is calculated using the standard Kaiser-Squires transformation, and compared to observations at the field level. The posterior distribution of maps given the shear data is sampled using Hamiltonian Monte Carlo chains. Our work improves on the original algorithm by making it numerically efficient, enabling full-sky reconstructions at $\approx$ 7 arcmin resolution with modest computational resources. These gains are made with no loss in accuracy or precision relative to KaRMMa 1.0. We compare the KaRMMa 2.0 posteriors against simulations across a variety of summary statistics (one-point function, two-point functions, and peak/void counts) to demonstrate our updated algorithm provides an accurate reconstruction of the convergence field at mildly non-linear scales. Unsurprisingly, the lognormal model fails as we approach non-linear scales ($\ell \gtrsim 200$), which in turn biases the map posteriors. These biases are at the 2% level in the recovered power spectrum, and at the 5% to 15% level for other statistics, depending on the resolution.