As an air rifle shooter I know there are always people looking for shooting permissions, most would never charge and certainly wouldn’t Come close to the cost of the large pest control companies if they did. maybe ask around your local community or explore on social media a bit if you use it?? I’m not saying it will irradiate your problem entirely (especially with unhelpful neighbours) but a few nights “sport” for someone semi regularly will certainly make a substantial dent.
Housing Association, there’ll not be any shooting allowed on the property.