Sorption heat storage (SHS) is a promising technology towards efficient use of renewable energy sources. Composite materials based on hygroscopic salts have a high potential for SHS in term of the heat storage capacity. Recently, a new sorbent “LiCl confined to Multi-Wall Carbon NanoTubes (MWCNT)” with enhanced storage capacity (1.7 kJ/g) has been suggested for SHS. This work addresses the dynamic study of water sorption on this material under operating conditions of a daily heat storage cycle. The study consists of three parts: (1) shaping the LiCl/MWCNT composite as grains (GP) and pellets with (PB) and without a binder (PP); (2) water sorption dynamics under conditions of the selected SHS cycle; (3) evaluation of the specific power achievable. Various configurations of the adsorbent bed were tested, namely, the loose grains GP placed on a metal support, and the pellets PB and PP glued to the support. The dynamic curves of water sorption obey an exponential equation. The PB configuration ensured a maximal specific power of 11.2 kW/kg (at 70% conversion) during the heat storage stage. During the heat release stage, the specific power is lower (5.4 kW/kg), probably due to a kinetic hindrance attributed to solid-solid transformation during the decomposition of LiCl·H2O complex towards the anhydrous salt. The high specific power along with the large heat storage capacity make the LiCl/MWCNT composite a promising new candidate for SHS.