Not sure if it would work, but you might be able to have the fade animation in the AnimationPlayer tell the AniamtedSprite that contains the balloon popping animation to play through a call method track.
You can make a call method track in an AnimationPlayer by opening the animation tab at the bottom of the Godot editor and then pressing the plus button in the top left of the panel. I believe the call method track is green. When you create the track it should have you point it to a node, which you'll want to point to the AnimatedSprite. Then once that is done, if you right click in the track at the keyframe position you want, you should be able to add a key and call the play function in the AnimatedSprite so it can play the balloon popping animation.
