I had to put my service dog of 15 years down in January. I know the heart-wrenching it causes to do this. What helped me was the advice of the veterinarian. When she said she just put her own dog down and he wasn't as bad as mine, that let me know that I was making the dog suffer just because I couldn't handle (or didn't want to) the loss of him. I still sob often over 'my' loss.

Maybe if you can convince your mom to take the dog to the vet for advice (medication or whatever) it will help?

She might be identifying with the "oldness" of the dog and herself and needs to know there's a big difference between aging parents and dogs.