LCC's actually do better when the cost of fuel goes higher. The legacies raise ticket prices to cover costs but since the spread between ticket price is how LCC's make money, with their CASM being half of the legacies, they make bank. Just go back in history over the past five years or so.

When oil prices fell the LCC's started to feel the squeeze from the legacies who were able to discount their tickets more to better compete.