Wonderfully, the official government webpage[1] lists his duties as:
Larry spends his days greeting guests to the house, inspecting security defences and testing antique furniture for napping quality. His day-to-day responsibilities also include contemplating a solution to the mouse occupancy of the house. Larry says this is still ‘in tactical planning stage’.
[1] https://www.gov.uk/government/history/10-downing-street#larr...