Thanks Sam. 4400kWh is a lot but the heat pump explains that if the hall has no other heating. I would say that 7.5kWh of battery is too small for both the proposed PV kWp and demand. On a sunny winter day (like yesterday Dec 19th) my neighbour with a south facing roof and 39degree roof angle generated 9.5 kWh from a 4kWp array. In summer he is generating in excess of 20kWh a day.
You not mention the size of the inverter / charger for the battery, but I guess it is 3kW? Also the size of the Solar inverter (4kW)?
If possible I would double the battery, and ensure the system had 6 or even 9kW of inverter / power supply for the them. In winter, you would have the option to store all the electricity you could from the roof to make it last as long as possible between sunny days - ideally being able to draw on the stored capacity without having to also import from the grid - would minimise the grid bill. In summer, having the potential to fully run any heating needed (water or space) from the batteries without also drawing from the grid, potentially for a couple of days would minimise grid cost. I would say payback is 10 - 15 years not including rising electricity costs, and not including revenue from selling excess capacity back to the grid.