Sparse measurements of glacier mass balance, velocity and ice thickness in Patagonia challenge our ability to understand glacier sensitivity to climate change and relate past glacier fluctuations to palaeoclimate change. Small ice caps, such as Monte San Lorenzo, have short response times and high climate sensitivity, making well-dated moraines in their glacier foregrounds an important tool for exploring glacier response to rapid changes in palaeoclimate. Here, the Parallel Ice Sheet Model (PISM) is used to model ice flow across a domain centred on the Monte San Lorenzo ice cap. Ice-flow parameters are calibrated to match present-day ice extent, velocity and thickness. Our aim is, firstly, to quantify present-day physical glacier properties, and ice cap dynamics and sensitivities, and secondarily, to evaluate the controls on the deglaciation of the ice cap within the context of the Southern Hemisphere palaeoclimate system during the Last Glacial-Interglacial Transition (LGIT). The simulated present-day ice cap shows high surface mass flux, with ablation at outlet glacier tongues up to 18 m w. e. a−1, accumulation at the highest elevations of up to 5.5 m w. e. a−1 and a simulated Equilibrium Line Altitude (ELA) of 1750–2000 m asl. The ice cap is more sensitive to changes in precipitation relative to changes in temperature. We provide envelopes with likely ranges of palaeotemperature and palaeoprecipitation for glacial advances to moraines formed during the Last Glacial-Interglacial Transition and Holocene. Our numerical model predicts that cooling and an increase in precipitation is required to force glacial advance to mapped moraine limits at 12.1 ka (2°C cooler, 50% more precipitation), 5.6 ka (0°C cooler, 50% more precipitation) and 0.2 ka (1°C cooler, 25% more precipitation). Our modelling results thus provide insights into the present-day mass balance, thermal regime and velocity of the ice cap, explores the sensitivities of this ice cap to various model and climatic parameters, and provide palaeoclimatic envelopes for readvances during the LGIT and Holocene in Patagonia.