You need to do enough damage to bring the green bar down which will then bring your champion back. 

Only way is to have them die before they get eaten.

This is it.

I try to preplan in the team building stages to ensure there is a method any combination of my 5 remaining team members can get enough damage even onto Mischief to free the champ eaten. Making sure I have enough AoE, burn, and brimstone damage really prevents a champ being swallowed and needing to restart a run.

I have a Duchess and Pythion on each team for revive.  They dont stay dead for long.  I wondered if maybe there was a secret method I had missed.  lol

Back to better gear.