Actually, that paragon of DU grammarians got it wrong as well. It should be "Barney and me".
Unless you're the President and you're talking to a little girl who's been traumatized by union thugs. Then you can say it any way you want.
I am a union Steamfitter and I just have to say that most painters are okay, but there are some that are just nutty.
(too much huffing?)
I think the fact that Barney is in fact a DOG, "Me and Barney" is justified in this case.