Was your dog ok before you put your pet rat down?
Were they friends? Sometimes animals will get attached to one another. Just guessing here. I saw a show about unusual pets being friends. When they were seperated, they went into mourning. One case was a goat and a donkey. The goat would not even get up to eat. And just lay there. Till they figured it out and got the donkey there.
Did it hear the gunshot? Some animals fall apart hearing one.
If neither is
the answer, I would most definately take it to the vet.
I think I would anyway, just to be safe.
I hope your dog is ok. Let us know ok?