Even though it's unfair, if the choice is "the burdens fall on the poor" and "the burdens fall on everyone (including the poor, who will be hit harder because they have fewer resources in reserve to survive unplanned catastrophe)", the first option is better.
(... it's also a false forced choice; there are other options. But the question of poverty runs somewhat separate to the question of saving humanity from climate change).
It is if there will be civil unrest, leading to the election of reactionary populist governments which roll back emission cuts.
Approaches to the problem that enrage a large portion of the citizenry are counterproductive. They won't work in a democracy, and most of the large polluters are democracies.
The French people are generally considered to be pretty left-wing (at least by outsiders) but they took to the streets as soon as the government raised fuel prices. Imagine how well that kind of policy would go down in Texas.
Poverty is therefore directly related to the question of solving climate change.
Therefore, market to the people climate-friendly lifestyles. Show celebrities doing their part. Make it easy to pick the right option by default with purchases. Encourage more local events and travel. It has been demonstrated that people respond to good storytelling and I don't think that is going to change.
The French suffered higher overhead energy costs to make their electricity production >70% nuclear. The difference is that this long term suffering is much less severe than short-term suffering.
It is a difficult problem to solve for a reason.
People are willing to suffer short-term privations, especially when they are afraid of getting the coronavirus. People are not willing to suffer for years to reduce their carbon footprints.
It's also notable that the burdens of reducing carbon emissions fall disproportionately on the poor. The fuel taxes in France are a perfect example.