My W and I still share a dog as well. If anything happened to it, I would tell her all about it. If she cares about the dog (and some people care very much about their pets) it would be overly harsh to deny her that information.
She is asking about how the dog is, so I would tell her what was wrong and what the vet said. You can still keep it brief and business like.
W 34 Me 42 Married 7 years together 8 0 kids 1 beloved dog BD 4/6/2018 I moved out 4/7/2018 I moved back in alone 8/05/2018 I file 3/06/2019 D official 5/7/2019