Animals transported Deaths Rate (deaths per 10,000)
United 344,483 41 1.19
Delta 235,179 18 0.77
American 210,216 9 0.43
SkyWest 123,612 3 0.24
Alaska 330,911 7 0.21

In fairness, the airline in recent years has also been the largest transporter of animals -- mostly dogs and cats, but also more exotic pets such as birds and geckos. Also, according to the DOT report for 2017, several of the animals had pre-existing health issues such as heart disease. (The report did not specify how many animals were flown in the cargo hold as opposed to the plane's cabin.)
But even when accounting for their larger volume of animal passengers, United still leads all other airlines in rate of deaths per 10,000 pets.
United did not respond to multiple requests for comment for this story. But spokesman Charlie Hobart told CNN on Tuesday that the airline is investigating the French bulldog's death "to prevent this from ever happening again."
As for the 2017 report, he said, "The overwhelming majority of the incidents were attributed to animals not being acclimated to its crate or the animal having a pre-existing condition we weren't aware of. Any time there is an incident with animals we do a thorough review."