I think increasing the payroll tax is a mistake. People who don't own a car should actually be rewarded because they pollute far less, they don't disable workers, they don't require traffic police, they don't emit CO2....
I would replace the payroll tax with a weight tax on huge SUVs.
There is tons of stats at data.bts.gov, BLS, and Federal reserve FRED system. Image above is from https://data.bts.gov/stories/s/Transportation-Economic-Trends-Transportation-Spen/ida7-k95k/
Basically, what we find is that wealthier people have bigger carbon footprint. They drive more miles, own more and bigger cars. They also fly more miles. What you are calling "degrees of obligation" is nothing more than a lifestyle choice. The suburbanite driving 50 miles a day in a BMW SUV is the one being impacted here, not the low-income worker taking the bus or driving an old Corolla.
Also note that driving is highly subsidized, and if the gas tax isn't raised to cover those costs then that money still has to come from somewhere. And that somewhere is other government programs, which low-income are much more highly dependent on.