I wonder if, instead of a large, hard-to-conceive-of number, you could, say, go with a game statistic. Call it Fate, call it kismet, call it "plot armour", call it whatever you like.
Each player character begins with a certain amount of Fate; in honour of cats, let's say 8, so that you have "nine lives" - your starting "life" plus eight more from your Fate. But you could make the number as big or as small as you like.
When a player character reaches 0 hit points, they don't go through the death saves business while unconscious. Instead, they lose 1 Fate and are restored immediately to full hit points.
If a player character reaches 0 hit points while they have 0 Fate, that's it: they die.