I don't know about training the dog, but you can attach a cage on the inside of the letter box to catch the mail, or as we do at our house, a letter box attached to the outside. that should do it.
I've got a wire cage on the inside, but the dogs can get the corners of letters if they are determined, or if the free paper gets shoved through and doesn't drop into the cage.
I don't think they are after the post as such, they are seeing off that horrible man who keeps coming back every day onto their property (you'd have thought he'd have learnt by now).
Know just what you mean, we have a 26kg staff who thinks whatever comes through the letterbox is fair game. The postie says its like Arkrights till. We put a L/B on the wall with a message printed on the flap which says,
"USE ME OR LOOSE EM"