
It is the environmental health part of the council that deals with noise complaints. The way it works is when they receive a complaint, they send a letter out to the dog owner, and another to the person complaining. The complainer has to keep a diary for 2 weeks detailing all times they have heard barking and for how long. They then send it back to the council. Quite possibly full of lies -my ex neighbours stated my dogs barked non stop for 3 hours every evening between 7 pm and 10 pm! (Would have been hard for me to watch EastEnders if they did!!) Then the council look at the diary, and if it looks like there is too much barking ( bearing in mind that dogs ARE by law allowed to bark, just not for too prolonged periods a time or at anti-social hours) they ask the complainer if they want listening equipment to be installed. They can only refuse so many times, if they don't agree they will not be allowed to make further complaints. If they install listening equipment you will not be told, and it will be on your neighbour's property. The council will then have a meeting with several people all listening to the recordings, and then they decide if there is a problem or not. If there isn't, they don't tell the dog owner a thing (very annoying) but they tell the complainer. If there is, they will tell the dog owner and give them a chance to do something to stop it.
I had EIGHT letters of complaints and twice they listened to the dogs. It was found each time to NOT be excessive barking. That's when the neighbours decided to ENSURE the barking was excessive and hence they started kicking our door etc to make the dogs bark

but that's another story. This is how it works anyway. The RSPCA have nothing to do with it, all they can do is say you leave your dog alone for too long -but by the sound of it you do not. However if you have a private landlord I'm not so sure that he can't do what he likes about his tenants i.e let them go if neighbours complain, whether there is a reason or not -I simply don't know as I have never rented a house. But in a nutshell I'd tell the neighbours to stop harassing you and if they have a problem make a formal complaint to the council and let THEM decide if there is a problem or not.
But yes, of course you also need to get your dog to be comfortable being left alone and not barking.
Good luck -I do know so well what its like and I wouldn't wish it on anyone.
Marianne. Dogs are not our whole lives, there are cats too!