Fuel cell based micro-combined heat and power (CHP) units used for domestic applications can provide significant cost and environmental benefits for end users and contribute to the UK's 2050 emissions target by reducing primary energy consumption in dwellings. Lately there has been increased interest in the development of systematic methods for the design of such systems and their smoother integration with domestic building services. Several models in the literature, whether they use a simulation or an optimisation approach, ignore the dwelling side of the system and optimise the efficiency or delivered power of the unit. However the design of the building services is linked to the choice of heating plant and its characteristics. Adding the dwelling's energy demand and temperature constraints in a model can produce more general results that can optimise the whole system, not only the micro-CHP unit. The fuel cell has various heat streams that can be harvested to satisfy heat demand in a dwelling and the design can vary depending on the proportion of heat needed from each heat stream to serve the energy demand. A mixed integer non-linear programming model (MINLP) that can handle multiple heat sources and demands is presented in this paper. The methodology utilises a process systems engineering approach. The model can provide a design that integrates the temperature and water flow constraints of a dwelling's heating system with the heat streams within the fuel cell processes while optimising total CO2 emissions. The model is demonstrated through different case studies that attempt to capture the variability of the housing stock. The predicted CO2 emissions reduction compared to a conventionally designed building vary from 27% to 30% and the optimum capacity of the fuel cell ranges between 1.9 kW and 3.6 kW. This research represents a significant step towards an integrated fuel cell micro-CHP and dwelling design.