the central point of your argument is flawed: the rich use orders of magnitude more carbon than the poor do.
further: even if it were true, the goal of a carbon tax is not to be progressive, but to disincentivize fossil fuel production, and to drive behavior toward less harmful alternatives.
This is the case for consumption taxes in general and carbon taxation in particular. This is not contradictory with the fact that the top income decile emits more per capita in absolute than the lowest one.
This is especially manifest for gasoline consumption: richer households consume significantly more of it, but it represents less than 2-3% of their income, while it can reach more than 10% for poorer households.
Indeed, this can and should be offset with a full redistribution of the tax proceeds - either flat or targeted at lower income households - to compensate the inherent regressivity of carbon taxation.
Source: my PhD was on the distributional consequences of carbon pricing.
a) poor people will end with more money than they paid
b) rich people will end with less money than they paid
c) no change in aggregate demand (although minor increase in velocity) so likely little additional inflation
So it still redistributed from rich to poor while disincentivizing carbon heavy processes.
This page (https://citizensclimatelobby.org/household-impact-study/) has a nice chart showing the fees and where they come from for an average household in each income quintile in the USA. The expected fees for each quintile (under a $15/ton carbon fee) are $173, $219, $252, $292, $381. On another page on the same site, you can find that the expected dividend (which just depends on the number of adults/children in the household) is $288.
If we take a flat percentage of the spending and give everyone the same fixed amount, poorer people will profit more than rich people.