Ideas
Open the cheat bar, type testingcheat on, then bb.showhiddenobjects.
Enter build and buy mode and type treat into the search bar, all the treats should show up. Grab yourself the wellness treat, OR you can buy a vet table in build and buy mode to use at home.
This has been my work-around!
I had this happen to me twice tonight, in both a modded save and an unmodded (and never modded previously) save. In both cases, not only did no vet staff ever turn up at the clinic, giving the animal a wellness treat failed to cure the illness. Their fur color would change back to normal, but instead of having the original illness they came in for, they would just start throwing up everywhere after being given the treat. And since the "event" was still going on, I couldn't leave the vet clinic because no staff had turned up to treat the pet. In the modded save, I used UI Cheats to cancel the list of event activities so the sim and dog could go home, but the dog was still vomiting when it got home. In the non-modded save, since UI Cheats wasn't installed, the sim and cat were trapped at the clinic, so I just exited the game without saving. (And lost about an hour of game play in the process.)