The size of the thermal storage for domestic hot water (DHW) production could be a limiting factor to the wide spread adoption of domestic heat pumps (HPs). To overcome this issue, a latent thermal energy storage (LTES) could represent a good solution, given its compactness and the independence from the cylindrical shape which characterizes sensible storages. Latent storage units are often built as fin-and-tube heat exchangers, with copper tubes and aluminum fins surrounded by a phase change material (PCM). Given that DHW production requires high power, such a heat exchanger (HX) is necessary to compensate for the low conductivity of the PCM, but increases the cost of the LTES. Increasing the thermal conductivity of the latent storage material would allow for the use of smaller and cheaper HX, with large benefits in term of costs, currently the main obstacle to a wider diffusion of this technology. In this work, the impact of a PCM with enhanced thermal conductivity on the HX design and cost is explored. A multi-objective optimization algorithm (TSEMO) is used to identify optimal configurations by varying HX design parameters and simulating the LTES for a complete charge/discharge cycle. The results demonstrate that improved thermal conductivity allows for a reduction in HX size and cost while maintaining performance. The best configurations achieved up to a 28.6% increase in efficiency compared to conventional SAT-based solutions.