i would use as a way to introduce her to death. eventually, a loved one will die, so a pet, however upsetting it may be, might be a good way open up communication about death and what you want teach her about your beliefs of the afterlife or the whys in life.
our cat had to be put to sleep when my daughter was just 3, she was told that she got very sick and had to go to heaven. now at almost 6, she still refers to the cat as being in heaven, and it makes it easier for her to ask questions about when we die. we have her buried with a rock.
i would not promote the new puppy too much. what if that puppy got ill as well, that i believe is a lot for a child to deal with. i agree to get the puppy checked asap. also, its not a bad thing to stay away from replaceable living things. what i mean is, just because you have another puppy, it doesnt lessen the hurt of losing the other. its a rare case she happened to have 2 puppies, and i dont assume that would be the case again, so it would be best to teach her how to deal with loss without focusing on the other puppy since most losses wont have an "other", kwim.
i am sorry for your loss. i had many pets as a child and buried each one, even the fish. i had a beloved cat who died when i was 9, it was the first death of a pet. it might be nice to find a pic of the puppy and frame it, i actually had a scrapbook of the cat, and it really helped. good luck
ETA, just remembered when i lost my childhood cat, my aunt, a catlover, sent me a sympathy for the loss of your pet card. it meant a lot to think a grown up thought my cat dying was a big deal, not just a kid thing. have a relative that has pets send it to her in the mail right away, it really helped me realize even adults knew its hard to lose a pet.