Reduced complexity climate models are useful tools for quantifying decision‐relevant uncertainties, given their flexibility, computational efficiency, and suitability for large‐ensemble frameworks necessary for statistical estimation using resampling techniques (e.g., Markov chain Monte Carlo). Here we document a new version of the simple, open‐source, global climate model Hector, coupled with a 1‐D diffusive heat and energy balance model (Diffusion Ocean Energy balance CLIMate model) and a sea-level change module (Building blocks for Relevant Ice and Climate Knowledge) that also represents contributions from thermal expansion, glaciers and ice caps, and polar ice sheets. We apply a Bayesian calibration approach to quantify model uncertainties surrounding 39 model parameters with prescribed radiative forcing, using observational information from global surface temperature, thermal expansion, and other contributors to sea-level change. We find the addition of thermal expansion as an observational constraint sharpens inference for the upper tail of posterior equilibrium climate sensitivity estimates (the 97.5 percentile is tightened from 7.1 to 6.6 K), while other contributors to sea-level change play a lesser role. The thermal expansion constraint also has implications for probabilistic projections of global surface temperature (the 97.5 percentile for RCP8.5 2100 temperature decreases 0.3 K). Due to the model's parameterization of thermal expansion as an uncertain function of global ocean heat, we note a trade‐off between two ways of incorporating thermal expansion information: Ocean heat data provide a somewhat sharper equilibrium climate sensitivity estimate while thermal expansion data allow for constrained sea-level projections.