Hi Liz,
Keep up what you are doing. Be sure to have lots of water in it, so when he starts chewing it, he also gets water in his beak. Keep playing with it and him.
Also, place watery fruit, such as watermelon and canteloupe, maybe a little orange, in the travel carrier. Also, some veggies that may give a little water. This will also give him some hydration.
When I traveled with Merlin, I usually took a water bottle and took her out in the airport between airline flights to change the carrier and give her some additional water, in addition to all the watery foods that are placed in the carrier. But I'm assuming that you are talking about sending your grey as cargo. If you can take him in the cabin, it is a lot easier!
Blessings,
Maggie-