Would you file a complaint against a Lyft driver for doing this?

The driver is crazy to think that someone would put their cat in the trunk (unless it’s a hatchback, then it’s fine). I’ve never used Lyft, only Uber, so I don’t know if there’s an option to give special requests or instructions beforehand. I can understand why some people might not want a cat in their car, but he could have refused the ride. If he agrees to give you a ride, he should have just shut up. If he didn’t like it, he should have told you to find another ride. I wouldn’t report him unless he said something horrible, but I also wouldn’t tip him either. I don’t know if Lyft asks customers for reviews, but you could write a review about his rudeness. Other than that, leave it alone. If I were a ride share driver, I wouldn’t want animals in my car either. I like cats, but I would flat out refuse to allow a dog in my car. Lyft should find a way to let drivers know in advance if you have pets or lots of stuff. 

What's the problem ? I put my cat in the trunk on the way to the vets, it's safer there if there is an emergency stop.

Presume it was not for hours.

He should have reported YOU.

A Lyft driver doesn't have to let you take your cat in his car.  Be grateful he took you and your cat to the vet.  This is the lyft policy taken from their website. "Unless the passenger has a service animal, it's entirely up to the driver whether or not to allow the passenger's pet in the vehicle. We advise passengers to call their drivers after their ride request is accepted to confirm that it's OK to bring their pets."  

you can file a complaint , but lyft doesn't give a fu*k,so nothing will happen :(