Then you're in luck, you appear to be the lucky winner to have found a new and exciting use for the "Level Event" ability which Hipshot posted about. You could use that to trigger the animation, and then have the Level Event be an event which actually triggers the Toggle Immortality script inside your prefab. Its all on the composite skills post.
The only problem would be is that all of them would have their immortality triggered simultaneously, but this wouldn't be a problem so long as you didn't expect to have more than one of these crazy things on the screen at one time. If you were very dedicated to this I think the only way you could really make it work to have an easy to place mob, having multiple at once, each capable of being invincible at different times, would be to make a number of different prefabs, each working off a different event, and then make sure that when you place down groups no two of them work off the same event trigger in the same room.
If you wanted to you could also make a tower looking actor, make it look big and scary, and have it be the one which plays the animation and triggers the event, making all its minions invincible. This way it would seem more plausible when all its minions become unkillable simultaneously.