Wow...he is such a little baby, and I would want him to eat, too. I am not a Yorkie puppy expert, by any means, but I would worry less about the egg and skip cows milk. Egg protein is doggone near perfect so much so that all other proteins are measured by egg protein, and I think as he grows up, he will expand his food choices. I think goat milk might work for such a young baby...maybe even mix raw egg and goat milk with a little kibble blended to make a porridge for him. I would probably put some organic virgin coconut oil in it too and serve it to him warm and in small amounts...often.
I wish you the best. I bet he is much better off with you, since she was so apparently eager to send him on his way. If you can, post some pictures of your new tiny prince!