Liquid-liquid phase-separated (LLPS) aerosol particles are known to exhibit increased cloud condensation nuclei (CCN) activity compared to well-mixed ones due to a complex effect of low surface tension and non-ideal mixing. The relation between the two contributions as well as the molecular-scale mechanism of water uptake in the presence of an internal interface within the particle is to date not fully understood. Here we attempt to gain understanding in these aspects through steered molecular dynamics simulation studies of water uptake by a vapor-hydroxy-cis-pinonic acid-water double interfacial system at 200 and 300 K. Simulated free-energy profiles are used to map the water uptake mechanism and are separated into energetic and entropic contributions to highlight its main thermodynamic driving forces. Atmospheric implications are discussed in terms of gas-particle partitioning, intraparticle water redistribution timescales and water vapor equilibrium saturation ratios. Our simulations reveal a strongly temperature-dependent water uptake mechanism, whose most prominent features are determined by local extrema in conformational and orientational entropies near the organic-water interface. This results in a low core uptake coefficient (k(o/w)=0.03) and a concentration gradient of water in the organic shell at the higher temperature, while entropic effects are negligible at 200 K due to the association-entropic-term reduction in the free-energy profiles. The concentration gradient, which results from non-ideal mixing - and is a major factor in increasing LLPS CCN activity - is responsible for maintaining liquid-liquid phase separation and low surface tension even at very high relative humidities, thus reducing critical supersaturations. Thermodynamic driving forces are rationalized to be generalizable across different compositions. The conditions under which single uptake coefficients can be used to describe growth kinetics as a function of temperature in LLPS particles are described.