If we want to be carbon neutral in twenty years, poorer people need pretty strong incentives too. I don't see that working out if the carbon tax is mostly cost neutral for them.
They'll choose the $6 one, and keep their carbon tax dividend in their pocket too.
They won't really be worse off.
On the other hand, the rich will be worse off because they take more flights, consume more resources, and generally cause more carbon emissions anyway from their day-to-day lives.
Taking a car lane and/or parking and making a dedicated bus and bike lane out of it will reduce congestion and increase the mobility of nearly everyone.
However there are other things we can do as well to reduce emissions as well as increase life quality at the same time (improve public transport and bicycle infrastructure) where poor people benefit especially.
Granted, that won't get us to net zero, but there are a lot of high-impact, low investment changes we can all make.
Longer term, if fossil fuel energy places rise, I expect a lot more pressure put on local governments to make public transportation more accessible, and to remove legislation that blocks the building of walkable neighborhoods and cities. I'm convinced that the vast majority of driving we do is completely unnecessary, and is just the result of myopic laws about how we can build cities and suburbs that tend not to get challenged, which lead to us just building in a really dumb way that ensures everyone needs to drive.
They should receive a negative carbon tax for their good behavior, which could be delivered in a subsidy for the purchase of cleaner fuel sources.
In fact their lives might well get better if the rest of us are forced into using public transport.