What a horrible experience. Moving pets is stressful enough without having to deal with stuff like this. It looks like your credit card company let you down as well. I would try to find a way to appeal their decision. Contact the BBB and anybody else you think can help you. I've had decisions reversed after getting the BBB involved before.
All that being said, we transported animals ourselves, and we used used
https://islandpetmovers.com/ twice. I probably wouldn't do it myself again unless they went out of business. Because of their negotiated rates, I think it was actually cheaper to use them than when we moved the animals ourselves.