The fuel/carbon tax would still be behavior-shifting for low-income emitters because it would still apply to low-income emitters per marginal unit, and that part is likely overall regressive because fuel is a larger expenditures for low-incomes.

However, the part where the resulting revenue is pooled and payed out in an equal amount back per capita is progressive, since that payment is a greater fraction of a low income. Desirably, it also means that low-income people emitting less than the average would make money overall: consider a household consisting of a single mom and two kids that take public transit to work/school.