dogochondriac
n. A person who is excessively preoccupied by their dog's health, or who tends to imagine ailments that the dog does not actually have.
Etymology
Examples
2013
Sooo… basically the vets say they can't see anything wrong with him, so maybe I'm just being a dogochondriac, but they do tend to miss things unless they are very obvious.
2011
If i had a dog, it would be a hypochondriac. A dogochondriac, for lack of a better word.
—alexd, “Re: Dr. Gabor Maté” (reply), TRIBE Main Forum, April 11, 2011
2004 (earliest)
Bonar, for one, learned pet CPR in a previous American Red Cross class. "It's similar to what you do with humans, but it is different, because a dog's anatomy is different."

But Bonar does not rely solely on her first-aid knowledge: "I'm a dogochondriac," she says. "If a dog even looks sick, I take it to the vet."
—Gary Pettus, “Learn first aid for pets,” The Clarion-Ledger, October 01, 2004
Notes
A shake of the paw to Fernando Montenegro for suggesting this word.