However, the first scenario will pass the increased cost of fuel down to the consumers affecting poorer people disproportionately. Example: some good that is produced with fossil fuels (including food) will become too expensive for low-income people, while richer (and more polluting) people will not feel the difference that much.

If you go for general taxation, you distribute the cost proportionally to income, making rich people pay more. Probably the ideal is a mix of both.