As Whiff said, there is a lot of money, time and expertise involved in keeping racing pigeons and whilst it is not as common as it used to be to see pigeon lofts, they will be very well kept and so will the birds. Where we lived before there was a retired civil servant who raced pigeons (so did HM the Queen, by the way), and I took him an injured ringed bird I had found on a walk in the woods. I was amazed at his set up, which spanned the width of his large garden. He even had a hospital wing! He showed me his stock books in the house, and from the ring on its leg he could tell that the bird had been flown from Basildon and was probably on the way back there when it was injured. He contacted the owner who subsequently came up here to South Yorkshire to collect his bird.
I would be very surprised if you were to have a problem with anyone keeping pigeons where you are proposing to live. The only time we have had a rat problem locally was from someone keeping chickens, not racing pigeons.