Hi A., first I am so sorry for the loss of your bird. Losing a pet is like losing a family member, so I understand that this is not an easy time.
I recently lost my cat, who died after being with me for 11 years. My 3 1/2 year old daughter was very attached to her. The cat used to sleep on her bed with her every night. They were thick as thieves!
We knew she was failing and prepared my daughter in advance, the same way that you have done with your kids. But we did not let her see the cat's dead body. We didn't want her to remember the cat that way, but instead wanted her only to think of the cat as the lively and fun pet that she knew and loved.
What we did instead was kind of what you said in your latter paragraph - we explained to her that the cat was too sick to stay with us anymore, and that sometimes when people or animals become that sick, only God can make them feel better. We told her that the cat had gone to live with God in Heaven, and that she was in a much better place feeling like her old self. She, of course, asked when the cat was going to be coming home - so we explained that once you go to live with God, you must stay with him because it is only in Heaven that you can be healthy once more.
It has proven to be a good lesson in death for her and has also given her a wonderful introduction to God. It was also the least traumatic way to explain the situation to her. She has coped with the loss well and we don't have to worry about how seeig the dead body might have affected her.
I hope this helps! And again, so sorry for your loss.