What an apt name for a moron who puts stupid replies.
Unfortunately whatever you put on the fence will just give them more things to perch on, the littlest birds can perch on a single stem. I have a similar problem, except for me its the birds making the dogs bark. However its a vicious circle because the dogs do scare away the birds. I suppose you could get a bird scarer like they use in fields, going off bang every couple of minutes, but that would probably drive you more mad than the birds. Ever thought about getting a cat.........